Robot modelling

Based on Section 2.8 of Robotics: Modelling, Planning and Control.

Link to the practical session on Robot Modelling

Denavit - Hartenberg Convention

Manipulator = series of links (rigid bodies) connected by joints (essentially prismatic or revolute), forming a kinematic chain, between the base and the end-effector.

The Denavit-Hartenberg convention is a method to derive the relative pose of two consecutive links of a kinematic chain, and is used to obtain a kinematic model of the manipulator.

../../_images/1.2.DH_convention.svg

Definition of frames

The frame \(i\) that describes the pose of link \(i\) is defined as follows:

  1. \(z_i\) axis: along joint \(i+1\).

  2. Origin \(O_i\) : at the intersection of \(z_i\) and the common normal to \(z_i, z_{i-1}\).

  3. \(x_i\) axis: along the common normal between \(z\) axis, from joint \(\: i\) towards joint \(i+1\), i.e., from \(z_{i-1}\) to \(z_{i}\).

  4. \(y_i\) axis: defined such that the frame is right-handed.

There are some special cases where the definitions are non-unique (choices are proposed to simplify the procedures).

  • Frame \(0: O_0\) and \(x_0\) are arbitrarily chosen, they may be set coincident with the manipulator base frame.

  • Frame \(n: z_n\) is not uniquely defined; it is usually chosen aligned with \(z_{n-1}\).

  • Two consecutive axes are parallel: there are infinite common normals, one of them is arbitrarily chose, usually the one that makes the \(d\) parameter to be zero (see below).

  • Two consecutive axes intersect: the sense of \(x_i\) is arbitrarily chosen, if possible in the same sense as \(x_{i-1}\).

  • Joint \(i\) is prismatic: the sense of \(z_{i-1}\) is arbitrarily chosen.

Definition of the transformation between frames

Then, according to the frames defined previously, the following parameters are introduced to define the transformation between link reference frames:

  • \(a_i\): distance between \(O_i'\) and \(O_i\)

  • \(d_i\): distance between \(O_{i-1}\) and \(O_i'\)

  • \(\alpha_i\): angle between \(z_{i-1}\) and \(z_i\) (positive = counter-clockwise)

  • \(\theta_i\): angle between \(x_{i-1}\) and \(x_i\) about \(z_{i-1}\) axis

Note that:

  • \(a_i, \alpha_i\) = constant values always, depend on the geometry only.

  • \(\theta_i\) = variable for revolute joints.

  • \(d_i\) = variable for prismatic joints.

Finally, the transformation between frames is computed through the intermediate frame \(O_i'\!\!-\!x_i'\,y_i'\,z_i'\):

\[\begin{split}\boldsymbol{A}_{i'}^{i-1} = \begin{bmatrix} c\theta_i & -s\theta_i & 0 & 0 \\ s\theta_i & c\theta_i & 0 & 0 \\ 0 & 0 & 1 & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad \text{Starting from frame } i-1\text{, translate } d_i \text{ along } z_{i-1}\text{, and rotate about } z_{i-1} \text{ an angle } \theta_i.\end{split}\]
\[\begin{split}\boldsymbol{A}_{i}^{i'} = \begin{bmatrix} 1 & 0 & 0 & a_i \\ 0 & c\alpha_i & -s\alpha_i & 0 \\ 0 & s\alpha_i & c\alpha_i & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \quad \text{Translate } a_i\text{ along } x_i' \text{ and rotate it about } x_i' \text{ an angle } \alpha_i.\end{split}\]
\[\begin{split}\boldsymbol{A}_{i}^{i-1} = \boldsymbol{A}_{i'}^{i-1} \boldsymbol{A}_{i}^{i'} = \begin{bmatrix} c\theta_i & -s\theta_i c\alpha_i & s\theta_i s\alpha_i & a_i c\theta_i \\ s\theta_i & c\theta_i c\alpha_i & -c\theta_i s\alpha_i & a_i s\theta_i \\ 0 & s\alpha_i & c\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

Procedure

The following procedure should be followed to determine the DH parameters of a manipulator:

Intitialize:

  1. Find and number consecutively the joint axes: \(z_0, ... z_{n-1}\)

  2. Choose frame 0 by locating the origin on \(z_0\); choose \(x_0\) arbitrarily (make it coincident with that of the base frame, if possible) and select \(y_0\) in order to have a right-handed frame.

Iterate: Repeat steps 3 to 5 for \(i=1\) to \(n-1\).

  1. Locate \(O_i\) at the intersection of \(z_i\) with the common normal \((z_{i-1},z_i)\).

However, if \(z_{i-1}\) and \(z_i\) are parallel:

\[\begin{split}\begin{cases} \text{locate } O_i \text{ such that } d_i = 0 \text{ (when joint is revolute),}\\ \text{locate } O_i \text{ at the mechanical limit of the joint (when joint is prismatic).} \end{cases}\end{split}\]

  1. Choose \(x_i\) along the common normal \((z_{i-1},z_i)\) from \(z_{i-1}\) to \(z_i\).

  2. Choose \(y_i\) to be right-handed.

Finalize:

  1. Choose Frame \(n\) such that:

\[\begin{split}\begin{cases} z_n \text{ and } z_{n-1} \text{ are aligned (revolute joint } n \text{).} \\ z_n \text{ arbitrary (prismatic joint } n \text{).} \end{cases}\end{split}\]
\[x_n \text{ is chosen as in step 4. }\]

  1. Write table of parameters \(a_i, d_i, \alpha_i, \theta_i\).

The following videos summarize the reference frames used for the definition of the DH parameters used to model robotic manipulators:

../../_images/video1.png

Link video 1

../../_images/video2.png

Link Video 2

Modified Denavit - Hartenberg Convention

This is a variant of the DH parameters, introduced by Craig in 1986, that choses \(z_i\) to be along joint \(i\) axis:

../../_images/modified_DH.svg
\[\begin{split}\boldsymbol{A}_{{i-1}'}^{i-1} = \begin{bmatrix} 1 & 0 & 0 & a_{i-1} \\ 0 & c\alpha_{i-1} & -s\alpha_{i-1} & 0 \\ 0 & s\alpha_{i-1} & c\alpha_{i-1} & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]
\[\begin{split}\boldsymbol{A}_{i}^{{i-1}'} = \begin{bmatrix} c\theta_i & -s\theta_i & 0 & 0 \\ s\theta_i & c\theta_i & 0 & 0 \\ 0 & 0 & 1 & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]
\[\begin{split}\boldsymbol{A}_{i}^{i-1} = \boldsymbol{A}_{{i-1}'}^{i-1} \boldsymbol{A}_{i}^{{i-1}'} = \begin{bmatrix} c\theta_i & -s\theta_i & 0 & a_{i-1} \\ s\theta_i c\alpha_{i-1} & c\theta_i c\alpha_{i-1} & -s\alpha_{i-1} & -s\alpha_{i-1} d_i \\ s\theta_i s\alpha_{i-1} & c\theta_i s\alpha_{i-1} & c\alpha_{i-1} & c\alpha_{i-1} d_i \\ 0 & 0 & 0 & 1 \end{bmatrix}\end{split}\]

Unified Robotic Description Format (URDF)

This is an alternative way to model kinematic manipulators. It provides a flexible method to define the transformations between links using six degrees of freedom.

../../_images/URDF.svg

origin = Transformation that locates the child link w.r.t the parent link.

\[\begin{split}\begin{cases} \text{position XYZ} \\ \text{ RPY angles} \end{cases}\end{split}\]

axis = joint-axis expressed in the child frame (defaults to \([1\;0\;0]^{T}\) ).

Examples

In the Direct Kinematics section (2.1) we will derive the DH parameters of:

  • the three-link planar arm

  • the spherical arm

  • the anthropomorphic arm

  • the spherical wrist