Denavit–Hartenberg parameters

A commonly used convention for selecting frames of reference in robotics applications is the Denavit and Hartenberg (D–H) convention which was introduced by Jacques Denavit and Richard S. Hartenberg. In this convention, coordinate frames are attached to the joints between two links such that one transformation is associated with the joint {{math|[Z ]}}, and the second is associated with the link {{math|[X ]}}. The coordinate transformations along a serial robot consisting of {{mvar|n}} links form the kinematics equations of the robot:

:$[T]\; =\; [Z\_1][X\_1][Z\_2][X\_2]\backslash ldots[X\_\{n-1\}][Z\_n][X\_n]\backslash !$

where {{math|[T ]}} is the transformation that characterizes the location and orientation of the end-link.

To determine the coordinate transformations {{math|[Z ]}} and {{math|[X ]}}, the joints connecting the links are modeled as either hinged or sliding joints, each of which has a unique line {{mvar|S}} in space that forms the joint axis and define the relative movement of the two links. A typical serial robot is characterized by a sequence of six lines {{math|1=S{{sub|i}} (i = 1, 2, ..., 6)}}, one for each joint in the robot. For each sequence of lines {{mvar|S{{sub|i}}}} and {{math|S{{sub|i+1}}}}, there is a common normal line {{math|A{{sub|i,i+1}}}}. The system of six joint axes {{mvar|S{{sub|i}}}} and five common normal lines {{math|A{{sub|i,i+1}}}} form the kinematic skeleton of the typical six degree-of-freedom serial robot. Denavit and Hartenberg introduced the convention that z-coordinate axes are assigned to the joint axes {{mvar|S{{sub|i}}}} and x-coordinate axes are assigned to the common normals {{math|A{{sub|i,i+1}}}}.

This convention allows the definition of the movement of links around a common joint axis {{mvar|S{{sub|i}}}} by the screw displacement:

:

where {{mvar|θ{{sub|i}}}} is the rotation around and {{mvar|d{{sub|i}}}} is the sliding motion along the {{mvar|z}}-axis. Each of these parameters could be a constant depending on the structure of the robot. Under this convention the dimensions of each link in the serial chain are defined by the screw displacement around the common normal {{math|A{{sub|i,i+1}}}} from the joint {{math|S{{sub|i}}}} to {{math|S{{sub|i+1}}}}, which is given by

:

where {{math|α{{sub|i,i+1}}}} and {{math|r{{sub|i,i+1}}}} define the physical dimensions of the link in terms of the angle measured around and distance measured along the X axis.

In summary, the reference frames are laid out as follows:

1. The {{mvar|z}}-axis is in the direction of the joint axis.
2. The {{mvar|x}}-axis is parallel to the common normal: $x\_n\; =\; z\_n\; \backslash times\; z\_\{n-1\}$ (or away from {{math|z{{sub|n–1}}}})
   If there is no unique common normal (parallel {{mvar|z}} axes), then {{mvar|d}} (below) is a free parameter. The direction of {{mvar|x{{sub|n}}}} is from {{math|z{{sub|n–1}}}} to {{mvar|z{{sub|n}}}}, as shown in the video below.
3. the {{mvar|y}}-axis follows from the {{mvar|x}}- and {{mvar|z}}-axes by choosing it to be a right-handed coordinate system.

File:Classic DH Parameters Convention.png

The following four transformation parameters are known as D–H parameters:{{cite book|last1=Spong|first1=Mark W.|last2=Vidyasagar|first2=M.|title=Robot Dynamics and Control |publisher=John Wiley & Sons|location=New York|year=1989|isbn= 9780471503521}}

• {{mvar|d}}: offset along previous {{mvar|z}} to the common normal
• {{mvar|θ}}: angle about previous {{mvar|z}} from old {{mvar|x}} to new {{mvar|x}}
• {{mvar|r}}: length of the common normal (aka {{mvar|a}}, but if using this notation, do not confuse with {{mvar|α}}). Assuming a revolute joint, this is the radius about previous {{mvar|z}}.
• {{mvar|α}}: angle about common normal, from old {{mvar|z}} axis to new {{mvar|z}} axis

There is some choice in frame layout as to whether the previous {{mvar|x}} axis or the next {{mvar|x}} points along the common normal. The latter system allows branching chains more efficiently, as multiple frames can all point away from their common ancestor, but in the alternative layout the ancestor can only point toward one successor. Thus the commonly used notation places each down-chain {{mvar|x}} axis collinear with the common normal, yielding the transformation calculations shown below.

We can note constraints on the relationships between the axes:

• the {{mvar|x{{sub|n}}}} axis is perpendicular to both the {{math|z{{sub|n–1}}}} and {{mvar|z{{sub|n}}}} axes
• the {{mvar|x{{sub|n}}}} axis intersects both {{math|z{{sub|n–1}}}} and {{mvar|z{{sub|n}}}} axes
• the origin of joint {{mvar|n}} is at the intersection of {{mvar|x{{sub|n}}}} and {{mvar|z{{sub|n}}}}
• {{mvar|y{{sub|n}}}} completes a right-handed reference frame based on {{mvar|x{{sub|n}}}} and {{mvar|z{{sub|n}}}}

It is common to separate a screw displacement into product of a pure translation along a line and a pure rotation about the line,{{cite journal|last1=Legnani|first1=Giovanni|last2=Casolo|first2=Federico|last3=Righettini|first3=Paolo|last4=Zappa|first4=Bruno|title=A homogeneous matrix approach to 3D kinematics and dynamics — I. Theory|journal=Mechanism and Machine Theory|volume=31|issue=5|year=1996|pages=573–587|doi=10.1016/0094-114X(95)00100-D}}{{cite journal|last1=Legnani|first1=Giovanni|last2=Casolo|first2=Federico|last3=Righettini|first3=Paolo|last4=Zappa|first4=Bruno|title=A homogeneous matrix approach to 3D kinematics and dynamics—II. Applications to chains of rigid bodies and serial manipulators|journal=Mechanism and Machine Theory|volume=31|issue=5|year=1996|pages=589–605|doi=10.1016/0094-114X(95)00101-4}} so that

:$[Z\_i]\; =\; \backslash operatorname\{Trans\}\_\{Z\_\{i\}\}(d\_i)\; \backslash operatorname\{Rot\}\_\{Z\_\{i\}\}(\backslash theta\_i),$

and

:$[X\_i]=\backslash operatorname\{Trans\}\_\{X\_i\}(r\_\{i,i+1\})\; \backslash operatorname\{Rot\}\_\{X\_i\}(\backslash alpha\_\{i,i+1\}).$

Using this notation, each link can be described by a coordinate transformation from the concurrent coordinate system to the previous coordinate system.

: $\{\}^\{n\; -\; 1\}T\_n$

= [Z_{n-1}]\cdot [X_n]

Note that this is the product of two screw displacements. The matrices associated with these operations are:

: $\backslash operatorname\{Trans\}\_\{z\_\{n\; -\; 1\}\}(d\_n)$

=

\left[

\begin{array}{ccc|c}

1 & 0 & 0 & 0 \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & d_n \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

: $\backslash operatorname\{Rot\}\_\{z\_\{n\; -\; 1\}\}(\backslash theta\_n)$

=

\left[

\begin{array}{ccc|c}

\cos\theta_n & -\sin\theta_n & 0 & 0 \\

\sin\theta_n & \cos\theta_n & 0 & 0 \\

0 & 0 & 1 & 0 \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

: $\backslash operatorname\{Trans\}\_\{x\_n\}(r\_n)$

=

\left[

\begin{array}{ccc|c}

1 & 0 & 0 & r_n \\

0 & 1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

: $\backslash operatorname\{Rot\}\_\{x\_n\}(\backslash alpha\_n)$

=

\left[

\begin{array}{ccc|c}

1 & 0 & 0 & 0 \\

0 & \cos\alpha_n & -\sin\alpha_n & 0 \\

0 & \sin\alpha_n & \cos\alpha_n & 0 \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

This gives:

: $\backslash operatorname\{\}^\{n\; -\; 1\}T\_n$

=

\left[

\begin{array}{ccc|c}

\cos\theta_n & -\sin\theta_n \cos\alpha_n & \sin\theta_n \sin\alpha_n & r_n \cos\theta_n \\

\sin\theta_n & \cos\theta_n \cos\alpha_n & -\cos\theta_n \sin\alpha_n & r_n \sin\theta_n \\

0 & \sin\alpha_n & \cos\alpha_n & d_n \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

=

\left[

\begin{array}{ccc|c}

& & & \\

& R & & T \\

& & & \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

where R is the 3×3 submatrix describing rotation and T is the 3×1 submatrix describing translation.

In some books, the order of transformation for a pair of consecutive rotation and translation (such as $d\_n$and $\backslash theta\_n$

) is reversed. This is possible (despite the fact that in general, matrix multiplication is not commutative) since translations and rotations are concerned with the same axes $z\_\{n-1\}$ and $x\_\{n\}$, respectively. As matrix multiplication order for these pairs does not matter, the result is the same. For example: $\backslash operatorname\{Trans\}\_\{z\_\{n\; -\; 1\}\}(d\_n)\; \backslash cdot$

\operatorname{Rot}_{z_{n - 1}}(\theta_n)

= \operatorname{Rot}_{z_{n - 1}}(\theta_n) \cdot

\operatorname{Trans}_{z_{n - 1}}(d_n)

.

Therefore, we can write the transformation $\backslash operatorname\{\}^\{n\; -\; 1\}T\_\{n\}$ as follows:

{}^{n - 1}T_n =

\operatorname{Trans}_{z_{n - 1}}(d_n) \cdot

\operatorname{Rot}_{z_{n - 1}}(\theta_n) \cdot

\operatorname{Trans}_{x_n}(r_n) \cdot

\operatorname{Rot}_{x_n}(\alpha_n)

{}^{n - 1}T_n =

\operatorname{Rot}_{z_{n - 1}}(\theta_n) \cdot

\operatorname{Trans}_{z_{n - 1}}(d_n) \cdot

\operatorname{Trans}_{x_n}(r_n) \cdot

\operatorname{Rot}_{x_n}(\alpha_n)

The Denavit and Hartenberg notation gives a standard (distal) methodology to write the kinematic equations of a manipulator. This is especially useful for serial manipulators where a matrix is used to represent the pose (position and orientation) of one body with respect to another.

The position of body $n$ with respect to $n-1$ may be represented by a position matrix indicated with the symbol $T$ or $M$

: $\backslash operatorname\{\}^\{n\; -\; 1\}T\_n\; =\; M\_\{n-1,n\}$

This matrix is also used to transform a point from frame $n$ to $n-1$

:

\hline

0 & 0 & 0 & 1 \end{array}\right]

Where the upper left $3\backslash times\; 3$ submatrix of $M$ represents the relative

orientation of the two bodies, and the upper right $3\backslash times\; 1$ represents their relative position or more specifically the body position in frame n − 1 represented with element of frame n.

The position of body $k$ with respect to body $i$ can be obtained as the product of the matrices representing the pose of $j$ with respect of $i$ and that of $k$ with respect of $j$

: $M\_\{i,k\}=\; M\_\{i,j\}\; M\_\{j,k\}$

An important property of Denavit and Hartenberg matrices is that the inverse is

:$M^\{-1\}\; =$

\left[

\begin{array}{ccc|c}

& & & \\

& R^T & & -R^T T \\

& & & \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

where $R^T$ is both the transpose and the inverse of the orthogonal matrix $R$, i.e. $R^\{-1\}\_\{ij\}=R^T\_\{ij\}\; =\; R\_\{ji\}$.

Further matrices can be defined to represent velocity and acceleration of bodies.

The velocity of body $i$ with respect to body $j$ can be represented in frame $k$ by the matrix

:

\hline

0 & 0 & 0 & 0 \end{array}\right]

where $\backslash omega$ is the angular velocity of body $j$ with respect to body $i$ and all the components are expressed in frame $k$; $v$ is the velocity of one point of body $j$ with respect to body $i$

(the pole). The pole is the point of $j$ passing through the origin of frame $i$.

The acceleration matrix can be defined as the sum of the time derivative of the velocity plus the velocity squared

:$H\_\{i,j(k)\}=\backslash dot\{W\}\_\{i,j(k)\}+W\_\{i,j(k)\}^2$

The velocity and the acceleration in frame $i$ of a point of body $j$ can be evaluated as

:$\backslash dot\{P\}\; =\; W\_\{i,j\}\; P$

:$\backslash ddot\{P\}\; =\; H\_\{i,j\}\; P$

It is also possible to prove that

:$\backslash dot\{M\}\_\{i,j\}\; =\; W\_\{i,j(i)\}\; M\_\{i,j\}$

:$\backslash ddot\{M\}\_\{i,j\}\; =\; H\_\{i,j(i)\}\; M\_\{i,j\}$

Velocity and acceleration matrices add up according to the following rules

:$W\_\{i,k\}=\; W\_\{i,j\}\; +\; W\_\{j,k\}$

:$H\_\{i,k\}=\; H\_\{i,j\}\; +\; H\_\{j,k\}\; +\; 2W\_\{i,j\}\; W\_\{j,k\}$

in other words the absolute velocity is the sum of the parent velocity plus the relative velocity; for the acceleration the Coriolis' term is also present.

The components of velocity and acceleration matrices are expressed in an arbitrary frame $k$ and transform from one frame to another by the following rule

:$W\_\{(h)\}=M\_\{h,k\}\; W\_\{(k)\}\; M\_\{k,h\}$

:$H\_\{(h)\}=M\_\{h,k\}\; H\_\{(k)\}\; M\_\{k,h\}$

For the dynamics, three further matrices are necessary to describe the inertia $J$, the linear and angular momentum $\backslash Gamma$, and the forces and torques $\backslash Phi$ applied to a body.

Inertia $J$:

:

I_{yz} & y_g m \\ I_{zx} & I_{zy} & I_{zz} & z_g m \\

\hline

x_g m & y_g m & z_g m & m \end{array}\right]

where $m$ is the mass, $x\_g,\backslash ,\; y\_g,\backslash ,\; z\_g$ represent the position of the center of mass, and the terms $I\_\{xx\},\backslash ,I\_\{xy\},\backslash ldots$ represent inertia and are defined as

:$I\_\{xx\}\; =\backslash iint\; x^2\; \backslash ,\; dm$

: $$

\begin{align}

I_{xy} & =\iint xy \, dm \\

I_{xz} & = \cdots \\

& \,\,\, \vdots

\end{align}

Action matrix $\backslash Phi$, containing force $f$ and torque $t$:

:

\hline

-f_x & -f_y & -f_z & 0 \end{array}\right]

Momentum matrix $\backslash Gamma$, containing linear $\backslash rho$ and angular $\backslash gamma$ momentum

:

\hline

-\rho_x & -\rho_y & -\rho_z & 0 \end{array}\right]

All the matrices are represented with the vector components in a certain frame $k$. Transformation of the components from frame $k$ to frame $h$ follows the rule

: $$

\begin{align}

J_{(h)} & = M_{h,k} J_{(k)} M_{h,k}^T \\

\Gamma_{(h)} & = M_{h,k} \Gamma_{(k)} M_{h,k}^T \\

\Phi_{(h)} & = M_{h,k} \Phi_{(k)} M_{h,k}^T

\end{align}

The matrices described allow the writing of the dynamic equations in a concise way.

Newton's law:

: $\backslash Phi\; =\; H\; J\; -\; J\; H^t\; \backslash ,$

Momentum:

: $\backslash Gamma\; =\; W\; J\; -\; J\; W^t\; \backslash ,$

The first of these equations express the Newton's law and is the equivalent of the vector equation $f\; =\; m\; a$ (force equal mass times acceleration) plus $t\; =\; J\; \backslash dot\{\backslash omega\}\; +\; \backslash omega\; \backslash times\; J\; \backslash omega$ (angular acceleration in function of inertia and angular velocity); the second equation permits the evaluation of the linear and angular momentum when velocity and inertia are known.

Some books such as Introduction to Robotics: Mechanics and Control (3rd Edition) John J. Craig, Introduction to Robotics: Mechanics and Control (3rd Edition) {{ISBN|978-0201543612}}

use modified (proximal) DH parameters. The difference between the classic (distal) DH parameters and the modified DH parameters are the locations of the coordinates system attachment to the links and the order of the performed transformations.

File:DHParameter.png

Compared with the classic DH parameters, the coordinates of frame $O\_\{i-1\}$ is put on axis i − 1, not the axis i in classic DH convention. The coordinates of $O\_\{i\}$ is put on the axis i, not the axis i + 1 in classic DH convention.

Another difference is that according to the modified convention, the transform matrix is given by the following order of operations:

: $$

{}^{n - 1}T_n = \operatorname{Rot}_{x_{n-1}}(\alpha_{n-1}) \cdot \operatorname{Trans}_{x_{n-1}}(a_{n-1}) \cdot \operatorname{Rot}_{z_{n}}(\theta_n) \cdot \operatorname{Trans}_{z_{n}}(d_n)

Thus, the matrix of the modified DH parameters becomes

: $\backslash operatorname\{\}^\{n\; -\; 1\}T\_n$

=

\left[

\begin{array}{ccc|c}

\cos\theta_n & -\sin\theta_n & 0 & a_{n-1} \\

\sin\theta_n \cos\alpha_{n-1} & \cos\theta_n \cos\alpha_{n-1} & -\sin\alpha_{n-1} & -d_n \sin\alpha_{n-1} \\

\sin\theta_n\sin\alpha_{n-1} & \cos\theta_n \sin\alpha_{n-1} & \cos\alpha_{n-1} & d_n \cos\alpha_{n-1} \\

\hline

0 & 0 & 0 & 1

\end{array}

\right]

Note that some books (e.g.:{{cite book

| last1 = Khalil

| first1 = Wisama

| last2 = Dombre

| first2 = Etienne

| title = Modeling, identification and control of robots

| publisher = Taylor Francis

| location = New York

| year = 2002

| url = https://books.google.com/books?id=Nx4X95PNyAkC

| isbn = 1-56032-983-1

| access-date = 2016-09-22

| archive-url = https://web.archive.org/web/20170312131028/https://books.google.com/books?id=Nx4X95PNyAkC

| archive-date = 2017-03-12

| url-status = live

}}) use $a\_n$ and $\backslash alpha\_n$ to indicate the length and twist of link n − 1 rather than link n. As a consequence, $\{\}^\{n\; -\; 1\}T\_n$ is formed only with parameters using the same subscript.

Surveys of DH conventions and its differences have been published.{{cite conference|last1=Lipkin|first1=Harvey|title=Volume 7: 29th Mechanisms and Robotics Conference, Parts a and B|volume=2005|year=2005|pages=921–926|doi=10.1115/DETC2005-85460|chapter=A Note on Denavit–Hartenberg Notation in Robotics|isbn=0-7918-4744-6}}{{cite book |last1=Waldron |first1=Kenneth |title=Springer Handbook of Robotics |last2=Schmiedeler |first2=James |year=2008 |pages=9–33 |doi=10.1007/978-3-540-30301-5_2 |chapter=Kinematics |isbn=978-3-540-23957-4}}

• Forward kinematics
• Inverse kinematics
• Kinematic chain
• Kinematics
• Robotics conventions
• Mechanical systems

{{commons category|Denavit-Hartenberg transformation}}

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Category:Robot control