End-effector pose and Denavit-Hartenberg frames

Pose of the end-effector

Given a fixed frame $𝑏$ attached to the base and another frame $𝑒$ attached to the end-effector, the pose of the latter is given by the homogeneous transformation matrix ${𝑇}_{𝑒}^{𝑏\left(𝒒\right)}$, where $𝒒$ is the $𝑛$-vector of joint variables.

${𝑛}_{𝑒}$, ${𝑠}_{𝑒}$, ${𝑎}_{𝑒}$ are the unit vectors associated with the end-effector and are chosen according to its geometry. For a two-fingered gripper, ${𝑎}_{𝑒}$ is chosen in the approach direction, ${𝑠}_{𝑒}$ in the sliding direction, and ${𝑛}_{𝑒}$ is orthogonal to the other two so that the frame is right-handed.

Open chains

In the general case, we have an open chain of $𝑛+1$ links connected by $𝑛$ joints (no loops between links).

By attaching a frame to each link, the pose of frame $𝑖+1$ is determined from the pose of frame $𝑖$, so finding the pose of the $𝑛$-th link is a recursive process starting from the base frame.

Denavit-Hartenberg convention

$𝑧$ axes are perpendicular to the viewer, passing through the joints

In order to find the direct kinematics of an open chain, we need an effective way to place the different frames.

The Denavit-Hartenberg convention proposes the following solution: frame $𝑖$ is attached to link $𝑖$, and its axis ${𝑧}_{𝑖}$ is chosen aligned with the axis of joint $𝑖+1$. We still have to choose ${\mathrm{𝟎}}_{𝑖}$ and the axes ${𝑥}_{𝑖}$ and ${𝑦}_{𝑖}$.

The common normal to two lines is the line containing the minimum-distance segment between them. It is uniquely defined when the lines are not coplanar; there is an infinite number of common normals when the lines are coplanar and parallel; and it degenerates to a single point when the lines are coplanar and intersect.

• The origin ${\mathrm{𝟎}}_{𝑖}$ is placed at the intersection between the common normal and the axis ${𝑧}_{𝑖}$, and ${\mathrm{𝟎}}_{{𝑖}^{\prime }}$ at its intersection with the axis ${𝑧}_{𝑖-1}$.
• ${𝑥}_{𝑖}$ is chosen along the common normal in the direction from ${𝑧}_{𝑖-1}$ to ${𝑧}_{𝑖}$.
• ${𝑦}_{𝑖}$ is chosen so that the frame is right-handed.

For frame $0$, only ${𝑧}_0$ is specified, so its origin and $𝑥$-axis can be arbitrary. For frame $𝑛$ there is no joint $𝑛+1$, so ${𝑧}_{𝑛}$ is not uniquely defined; since joint $𝑛$ is revolute, ${𝑧}_{𝑛}$ is aligned with ${𝑧}_{𝑛-1}$.

We introduce the following parameters:

• ${𝑎}_{𝑖}$: distance between ${\mathrm{𝟎}}_{𝑖}$ and ${\mathrm{𝟎}}_{{𝑖}^{\prime }}$;
• ${𝑑}_{𝑖}$: coordinate of ${\mathrm{𝟎}}_{{𝑖}^{\prime }}$ along ${𝑧}_{𝑖-1}$;
• ${𝛼}_{𝑖}$: rotation angle between ${𝑧}_{𝑖-1}$ and ${𝑧}_{𝑖}$ about ${𝑥}_{𝑖}$;
• ${𝜃}_{𝑖}$: rotation angle between ${𝑥}_{𝑖-1}$ and ${𝑥}_{𝑖}$ about ${𝑧}_{𝑖-1}$;

Construction of the transformation matrix

To build the transformation matrix between link $𝑖-1$ and link $𝑖$:

• We start from frame $𝑖-1$ and translate it by ${𝑑}_{𝑖}$ along ${𝑧}_{𝑖-1}$ and rotate it by ${𝜃}_{𝑖}$ about ${𝑧}_{𝑖-1}$, aligning it with frame ${𝑖}^{\prime }$.

  $${𝐴}_{{𝑖}^{\prime }}^{𝑖-1}=\left[\begin{array}{cccc}{𝑐}_{{𝜃}_{𝑖}}& -{𝑠}_{{𝜃}_{𝑖}}& 0& 0\\ {𝑠}_{{𝜃}_{𝑖}}& {𝑐}_{{𝜃}_{𝑖}}& 0& 0\\ 0& 0& 1& {𝑑}_{𝑖}\\ 0& 0& 0& 1\end{array}\right]$$

• Then we translate frame ${𝑖}^{\prime }$ by ${𝑎}_{𝑖}$ along ${𝑥}_{{𝑖}^{\prime }}$ and rotate it by ${𝛼}_{𝑖}$ about ${𝑥}_{{𝑖}^{\prime }}$, aligning it with frame $𝑖$.

  $${𝐴}_{𝑖}^{{𝑖}^{\prime }}=\left[\begin{array}{cccc}1& 0& 0& {𝑎}_{𝑖}\\ 0& {𝑐}_{{𝛼}_{𝑖}}& -{𝑠}_{{𝛼}_{𝑖}}& 0\\ 0& {𝑠}_{{𝛼}_{𝑖}}& {𝑐}_{{𝛼}_{𝑖}}& 0\\ 0& 0& 0& 1\end{array}\right]$$

The end result is:

$${𝐴}_{𝑖}^{𝑖-1}=\left[\begin{array}{cccc}{𝑐}_{{𝜃}_{𝑖}}& -{𝑠}_{{𝜃}_{𝑖}}{𝑐}_{{𝛼}_{𝑖}}& {𝑠}_{{𝜃}_{𝑖}}{𝑠}_{{𝛼}_{𝑖}}& {𝑎}_{𝑖}{𝑐}_{{𝜃}_{𝑖}}\\ {𝑠}_{{𝜃}_{𝑖}}& {𝑐}_{{𝜃}_{𝑖}}{𝑐}_{{𝛼}_{𝑖}}& -{𝑐}_{{𝜃}_{𝑖}}{𝑠}_{{𝛼}_{𝑖}}& {𝑎}_{𝑖}{𝑠}_{{𝜃}_{𝑖}}\\ 0& {𝑠}_{{𝛼}_{𝑖}}& {𝑐}_{{𝛼}_{𝑖}}& {𝑑}_{𝑖}\\ 0& 0& 0& 1\end{array}\right]$$