Denavit–Hartenberg parameters

Denavit–Hartenberg matrix

It is common to separate a screw displacement into product of a pure translation along a line and a pure rotation about the line, so that

${\displaystyle [Z_i]={\mathrm{Trans}}_{Z_i}(d_i){\mathrm{Rot}}_{Z_i}({\theta }_i),}$

and

${\displaystyle [X_i]={\mathrm{Trans}}_{X_i}(r_{i,i+1}){\mathrm{Rot}}_{X_i}({\alpha }_{i,i+1}).}$

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

${\displaystyle {}^{n-1}{T}_n=[Z_{n-1}]\cdot [X_n]}$

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

${\displaystyle {\mathrm{Trans}}_{z_{n-1}}(d_n)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& d_n\\ 0& 0& 0& 1\end{array}\right]}$

${\displaystyle {\mathrm{Rot}}_{z_{n-1}}({\theta }_n)=\left[\begin{array}{cccc}\cos {\theta }_n& -\sin {\theta }_n& 0& 0\\ \sin {\theta }_n& \cos {\theta }_n& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}$

${\displaystyle {\mathrm{Trans}}_{x_n}(r_n)=\left[\begin{array}{cccc}1& 0& 0& r_n\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}$

${\displaystyle {\mathrm{Rot}}_{x_n}({\alpha }_n)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\alpha }_n& -\sin {\alpha }_n& 0\\ 0& \sin {\alpha }_n& \cos {\alpha }_n& 0\\ 0& 0& 0& 1\end{array}\right]}$

This gives:

${\displaystyle {}^{n-1}{T}_n=\left[\begin{array}{cccc}\cos {\theta }_n& -\sin {\theta }_n\cos {\alpha }_n& \sin {\theta }_n\sin {\alpha }_n& r_n\cos {\theta }_n\\ \sin {\theta }_n& \cos {\theta }_n\cos {\alpha }_n& -\cos {\theta }_n\sin {\alpha }_n& r_n\sin {\theta }_n\\ 0& \sin {\alpha }_n& \cos {\alpha }_n& d_n\\ 0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cccc}& & & \\ & R& & T\\ & & & \\ 0& 0& 0& 1\end{array}\right]}$

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 ${\displaystyle d_n}$and ${\displaystyle {\theta }_n}$) 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 ${\displaystyle z_{n-1}}$ and ${\displaystyle x_n}$, respectively. As matrix multiplication order for these pairs does not matter, the result is the same. For example: ${\displaystyle {\mathrm{Trans}}_{z_{n-1}}(d_n)\cdot {\mathrm{Rot}}_{z_{n-1}}({\theta }_n)={\mathrm{Rot}}_{z_{n-1}}({\theta }_n)\cdot {\mathrm{Trans}}_{z_{n-1}}(d_n)}$.

Therefore, we can write the transformation ${\displaystyle {}^{n-1}{T}_n}$ as follows:
${\displaystyle {}^{n-1}{T}_n={\mathrm{Trans}}_{z_{n-1}}(d_n)\cdot {\mathrm{Rot}}_{z_{n-1}}({\theta }_n)\cdot {\mathrm{Trans}}_{x_n}(r_n)\cdot {\mathrm{Rot}}_{x_n}({\alpha }_n)}$
${\displaystyle {}^{n-1}{T}_n={\mathrm{Rot}}_{z_{n-1}}({\theta }_n)\cdot {\mathrm{Trans}}_{z_{n-1}}(d_n)\cdot {\mathrm{Trans}}_{x_n}(r_n)\cdot {\mathrm{Rot}}_{x_n}({\alpha }_n)}$

Use of Denavit and Hartenberg matrices

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 ${\displaystyle n}$ with respect to ${\displaystyle n-1}$ may be represented by a position matrix indicated with the symbol ${\displaystyle T}$ or ${\displaystyle M}$

${\displaystyle {}^{n-1}{T}_n=M_{n-1,n}}$

This matrix is also used to transform a point from frame ${\displaystyle n}$ to ${\displaystyle n-1}$

${\displaystyle M_{n-1,n}=\left[\begin{array}{cccc}{R}_{xx}& R_{xy}& R_{xz}& T_x\\ R_{yx}& R_{yy}& R_{yz}& T_y\\ R_{zx}& R_{zy}& R_{zz}& T_z\\ 0& 0& 0& 1\end{array}\right]}$

Where the upper left ${\displaystyle 3\times 3}$ submatrix of ${\displaystyle M}$ represents the relative orientation of the two bodies, and the upper right ${\displaystyle 3\times 1}$ represents their relative position or more specifically the body position in frame n − 1 represented with element of frame n.

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

${\displaystyle M_{i,k}=M_{i,j}{M}_{j,k}}$

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

${\displaystyle M^{-1}=\left[\begin{array}{cccc}& & & \\ & R^T& & -R^{T}T\\ & & & \\ 0& 0& 0& 1\end{array}\right]}$

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

Kinematics

Further matrices can be defined to represent velocity and acceleration of bodies. The velocity of body ${\displaystyle i}$ with respect to body ${\displaystyle j}$ can be represented in frame ${\displaystyle k}$ by the matrix

${\displaystyle W_{i,j(k)}=\left[\begin{array}{cccc}0& -{\omega }_z& {\omega }_y& v_x\\ {\omega }_z& 0& -{\omega }_x& v_y\\ -{\omega }_y& {\omega }_x& 0& v_z\\ 0& 0& 0& 0\end{array}\right]}$

where ${\displaystyle \omega }$ is the angular velocity of body ${\displaystyle j}$ with respect to body ${\displaystyle i}$ and all the components are expressed in frame ${\displaystyle k}$; ${\displaystyle v}$ is the velocity of one point of body ${\displaystyle j}$ with respect to body ${\displaystyle i}$ (the pole). The pole is the point of ${\displaystyle j}$ passing through the origin of frame ${\displaystyle i}$.

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

${\displaystyle H_{i,j(k)}={\dot{W}}_{i,j(k)}+W_{i,j(k)}^2}$

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

${\displaystyle \dot{P}=W_{i,j}P}$

${\displaystyle \ddot{P}=H_{i,j}P}$

It is also possible to prove that

${\displaystyle {\dot{M}}_{i,j}=W_{i,j(i)}{M}_{i,j}}$

${\displaystyle {\ddot{M}}_{i,j}=H_{i,j(i)}{M}_{i,j}}$

Velocity and acceleration matrices add up according to the following rules

${\displaystyle W_{i,k}=W_{i,j}+W_{j,k}}$

${\displaystyle H_{i,k}=H_{i,j}+H_{j,k}+2W_{i,j}{W}_{j,k}}$

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 ${\displaystyle k}$ and transform from one frame to another by the following rule

${\displaystyle W_{(h)}=M_{h,k}{W}_{(k)}{M}_{k,h}}$

${\displaystyle H_{(h)}=M_{h,k}{H}_{(k)}{M}_{k,h}}$

Dynamics

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

Inertia ${\displaystyle J}$:

${\displaystyle J=\left[\begin{array}{cccc}{I}_{xx}& I_{xy}& I_{xz}& x_{g}m\\ I_{yx}& I_{yy}& I_{yz}& y_{g}m\\ I_{zx}& I_{zy}& I_{zz}& z_{g}m\\ x_{g}m& y_{g}m& z_{g}m& m\end{array}\right]}$

where ${\displaystyle m}$ is the mass, ${\displaystyle x_g,y_g,z_g}$ represent the position of the center of mass, and the terms ${\displaystyle I_{xx},I_{xy},\dots }$ represent inertia and are defined as

${\displaystyle I_{xx}=\iint x^{2}dm}$

${\displaystyle \begin{array}{cc}{I}_{xy}& =\iint xydm\\ I_{xz}& =\cdots \\ & ⋮\end{array}}$

Action matrix ${\displaystyle \mathrm{\Phi }}$, containing force ${\displaystyle f}$ and torque ${\displaystyle t}$:

${\displaystyle \mathrm{\Phi }=\left[\begin{array}{cccc}0& -t_z& t_y& f_x\\ t_z& 0& -t_x& f_y\\ -t_y& t_x& 0& f_z\\ -f_x& -f_y& -f_z& 0\end{array}\right]}$

Momentum matrix ${\displaystyle \mathrm{\Gamma }}$, containing linear ${\displaystyle \rho }$ and angular ${\displaystyle \gamma }$ momentum

${\displaystyle \mathrm{\Gamma }=\left[\begin{array}{cccc}0& -{\gamma }_z& {\gamma }_y& {\rho }_x\\ {\gamma }_z& 0& -{\gamma }_x& {\rho }_y\\ -{\gamma }_y& {\gamma }_x& 0& {\rho }_z\\ -{\rho }_x& -{\rho }_y& -{\rho }_z& 0\end{array}\right]}$

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

${\displaystyle \begin{array}{cc}{J}_{(h)}& =M_{h,k}{J}_{(k)}{M}_{h,k}^T\\ {\mathrm{\Gamma }}_{(h)}& =M_{h,k}{\mathrm{\Gamma }}_{(k)}{M}_{h,k}^T\\ {\mathrm{\Phi }}_{(h)}& =M_{h,k}{\mathrm{\Phi }}_{(k)}{M}_{h,k}^T\end{array}}$

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

Newton's law:

${\displaystyle \mathrm{\Phi }=HJ-J{H}^t}$

Momentum:

${\displaystyle \mathrm{\Gamma }=WJ-J{W}^t}$

The first of these equations express the Newton's law and is the equivalent of the vector equation ${\displaystyle f=ma}$ (force equal mass times acceleration) plus ${\displaystyle t=J\dot{\omega }+\omega \times J\omega }$ (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.

Modified DH parameters

Some books such as Introduction to Robotics: Mechanics and Control (3rd Edition) 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.

Compared with the classic DH parameters, the coordinates of frame ${\displaystyle O_{i-1}}$ is put on axis i − 1, not the axis i in classic DH convention. The coordinates of ${\displaystyle O_i}$ 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:

${\displaystyle {}^{n-1}{T}_n={\mathrm{Rot}}_{x_{n-1}}({\alpha }_{n-1})\cdot {\mathrm{Trans}}_{x_{n-1}}(a_{n-1})\cdot {\mathrm{Rot}}_{z_n}({\theta }_n)\cdot {\mathrm{Trans}}_{z_n}(d_n)}$

Thus, the matrix of the modified DH parameters becomes

${\displaystyle {}^{n-1}{T}_n=\left[\begin{array}{cccc}\cos {\theta }_n& -\sin {\theta }_n& 0& a_{n-1}\\ \sin {\theta }_n\cos {\alpha }_{n-1}& \cos {\theta }_n\cos {\alpha }_{n-1}& -\sin {\alpha }_{n-1}& -d_n\sin {\alpha }_{n-1}\\ \sin {\theta }_n\sin {\alpha }_{n-1}& \cos {\theta }_n\sin {\alpha }_{n-1}& \cos {\alpha }_{n-1}& d_n\cos {\alpha }_{n-1}\\ 0& 0& 0& 1\end{array}\right]}$

Note that some books (e.g.:) use ${\displaystyle a_n}$ and ${\displaystyle {\alpha }_n}$ to indicate the length and twist of link n − 1 rather than link n. As a consequence, ${\displaystyle {}^{n-1}{T}_n}$ is formed only with parameters using the same subscript.

Surveys of DH conventions and its differences have been published.