Robot Dynamics & control: Lecture 3 - Forward Kinematics: The Denavit-Hartenberg Convention

Introduction

The forward kinematics problem is to determine the position and orientation of the end-effector, given the values for the joint variables of the robot.
The joint variables: angle between the links for revolute or rotational joint link extension for the prismatic or sliding joint.

$$ \begin{aligned} \begin{bmatrix} x \\ y \\ z \\ \psi \\ \theta \\ \phi \\ \end{bmatrix} = f(\theta_1, \cdots , \theta_n, d_1 , \cdots , d_n) \end{aligned} $$

$$f$$: forward kinematics

Kinematic Chains

A robot manupulator n joints will have n+1 links including ground.
Joint number: 1 ~ n, Link number: 0 ~ n
Joint i connects link i-1 to link i.
Joint i is fixed at link i-1, so when joint i is actuated, link i moves.
Link 0(the first link) is fixed and does not move when the joints are actuated.

With the i^{th} joint, we associate a joint variable, denoted by q_i.
- $$\theta_i$$: joint $$i$$ revolute
- $$d_i$$: joint $$i$$ prismatic
$$o_i x_i y_i z_i$$ is attached to link i $$\rightarrow$$ when joint i is actuated, link i and its attached frame $$o_i x_i y_i z_i$$ experience a resulting motion.
The frame $$o_0 x_0 y_0 z_0$$ which is attached to the robot base, is referred to as the inertial frame(=world coordinate).

Suppose $$A_i$$ is the homogeneous transformation matrix that expresses the position and orientation of $$o_i x_i y_i z_i$$ with respect to $$o_{i-1} x_{i-1} y_{i-1} z_{i-1}$$.
The matrix $$A_i$$ is not constant but varies as the configuration of the robot is changed, and $$A_i$$ is a function of only a single joint variable, namely $$q_i$$.

$$ \begin{aligned} A_i = A_i (q_i) \end{aligned} $$

Homogeneous transformation matrix $$T^i_j$$ that expresses the position and orientation of $$o_j x_j y_j z_j $$ with respect to $$o_i x_i y_i z_i$$ .

$$ \begin{aligned} T^i_j = A_{i+1} A_{i+2} \cdots A_{j-1} A_{j} \qquad if \quad i < j \\ T^i_j = I \qquad if \quad i = j \\ T^i_j = (T^j_i)^{-1} \qquad if \quad i > j \end{aligned} $$

The homogeneous transformation matrix denoted by the position($$o^0_n$$) and orientation ($$R^0_n$$) of the end-effector with respect to the inertial or base frame:
- This is forward kinematics!
- But, it is possible to achieve a considerable amount of stream linking and simplication by introducing D-H representation.

$$ \begin{aligned} H &= \begin{bmatrix} R^0_n & o^0_n\\ 0 & 1 \\ \end{bmatrix} \\ &= T^0_n \\ &= A_1(q_1) \cdots A_n(q_n) \end{aligned} $$

Each homogeneous transformation matrix:

$$ \begin{aligned} A_i &= \begin{bmatrix} R^{i-1}_i & o^{i-1}_i\\ 0 & 1 \\ \end{bmatrix} \end{aligned} $$

Hence,

$$ \begin{aligned} T^i_j &= A_{i+1} \cdots A_j &= \begin{bmatrix} R^{i}_j & o^{i}_j\\ 0 & 1 \\ \end{bmatrix} , R^i_{j} &= R^i_{i+1} \cdots R^{j-1}_j \\ o^i_{j} &= o^i_{j-1} + R^i_{j-1} o^{j-1}_j \end{aligned} $$

It is possible to carry out all of the analysis using an arbitrary frame attached to each link.
However, it is helpful to be systematic in the choice of these frames by using the Denavit-Hartenberg, or D-H convention.
In D-H convention, each homogeneous tranformation matrix $$A_i$$ is represented as a product of four basic transformations.

Denavit - Hartenberg Representation

$$ \begin{aligned} A &= Rot_{z, \theta_i} Trans_{z, d_i} Trans_{x, a_i} Rot_{x, \alpha_i} \\ &= \begin{bmatrix} c_{\theta_i} & -s_{\theta_i} & 0 & 0\\ s_{\theta_i} & c_{\theta_i} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & d_i\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & a_i\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & c_{\alpha_i} & - s_{\alpha_i}& 0\\ 0 & s_{\alpha_i} & c_{\alpha_i}& 0\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \\ &= \begin{bmatrix} c_{\theta_i} & -s_{\theta_i}c_{\alpha_i} & s_{\theta_i}s_{\alpha_i} & a_i c_{\theta_i}\\ s_{\theta_i} & c_{\theta_i}c_{\alpha_i} & -c_{\theta_i}s_{\alpha_i} & a_i s_{\theta_i}\\ 0 & s_{\alpha_i} & c_{\alpha_i}& d_i\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \end{aligned} $$

Since the matrix A_i is a function of a single variable, the 3 of the 4 quantities are constant for a given link.
- $$d_i$$: Joint variable for a prismatic joint.
- $$\theta_i$$: Joint variable for a revolute joint.
By a clever choice of the origin and coordinate axes, it is possible to cut down the number of parameters and needed from 6 to 4.
Existence and uniqueness issues:
- Clearly, it is not possible to represent any arbitrary homogeneous transformation using only four parameters
- But, is is possible to derive a unique homogeneous transformation matrix $$A$$ under the following two conditions.

DH1) The axis
$$x_1$$
is perpendicular to the axis
$$z_0$$
.
DH2) The axis
$$x_1$$
intersects the axis
$$z_0$$

Under DH1 and DH2, we claim that there exist unique numbers
$$a, d, \theta , \alpha$$
such that.
- $$A = Rot_{z,\theta} Trans_{z,d} Trans_{x,a} Rot_{x, \alpha}$$
Physical interpretation of four quantities:
- a: Distance between the axes $$z_0$$ and $$z_1$$ measured along the axis $$x_1$$
- d: Distance between the origin $$o_0$$ and the intersection of the $$x_1$$ axis with $$z_0$$ measured along the $$z_0$$ axis.

Physical interpretation of 4 quantities:
- $$\alpha$$: Angle between the axes $$z_0$$ and $$z_1$$ measured in a plane normal to $$x_1$$.
- $$\theta$$: Angle between $$x_0$$ and $$x_1$$ measured in a plane normal to $$z_0$$.

Assigning the coordinate frames
- For a given robot manipulator, one can always choose the frame 0, 1, …, n in such a way that the DH1 and DH2 are satisfied.
- It is important to keep in mind that the choices of the various coordinate frames are not unique, even when constrained by the DH1 and DH2.
- However, it is important to note that the end result($$T^0_n$$) will be the same, regardless of the assignment if untermediate link frames .
We assign
$$z_i$$
to be the axis of actuation for joint i+1
- If joint i+1 is revolute, $$z_i$$ is the axis of revolution of joint i+1.
- If joint i+1 is prismatic, $$z_i$$ is the axis of translation of joint i+1.

In order to set up frame i it is necessary to consider 3 cases:
case 1:
- $$z_{i-1}$$ and $$z_i$$ are not coplanar:

case 2:
- $$z_{i-1}$$ is parallel to $$z_i$$:
  - $$d_i, \alpha_i$$ are 0 in this case

case 3:
- $$z_{i-1}$$ intersects $$z_i$$:
  - $$a_i$$ are 0 in this case

This constructive procedure works frame 0, …, n-1 in an n-link robot.
The final coordinate system $$o_n x_n y_n z_n$$ is commonly referred to as the end-effctor(or tool frame)

Terminology arises from the fact that the direction a is the approach direction, the s direction is the sliding direction, and n is the direction normal to the plane formed by a and s.
Note the following important fact:
- The quantities $$a_i$$ and $$\alpha$$ are always constant (characteristics of the manipulator)
- If joint i is prismatic, then $$\theta_i$$ is also a constant, while $$d_i$$ is the $$i^{th}$$ joint variable.
- If joint i is revolute, then $$d_i$$ is also a constant, while $$\theta_i$$ is the $$i^{th}$$ joint variable.
Summary:

$$1.$$

Locate and label the joint axes

$$z_0, ..., z_{n-1}$$

$$2.$$

Establish the base frame. Set the origin anywhere on the

$$z_0$$

-axis. The

$$x_0$$

and

$$y_0$$

axed are chosen conveniently to form a right-hand frame. For i = 1, …, n-1, perform step 3 to 5.

$$3.$$

$$z_i$$

intersects

$$z_{i-1}$$

, locate

$$o_i$$

at this intersection. If

$$z_i$$

and

$$z_{i-1}$$

are parallel, locate

$$o_i$$

in any convenient position along

$$z_i$$

$$4.$$

Establish

$$x_i$$

along the common normal between

$$z_{i-1}$$

and

$$z_i$$

through

$$o_i$$

, or in the direction normal to the

$$z_{i-1} - z_i$$

plane if

$$z_{i-1}$$

and

$$z_i$$

intersect.

$$5.$$

Establish

$$y_i$$

to complete a right-hand frame

$$6.$$

Establish the end-effector frame

$$o_n x_n y_n z_n$$

. Assuming the n-th joint is revolute, set

$$z_n$$

= a along the direction

$$z_{n-1}$$

. Establish the origin on conveniently along

$$z_n$$

, preferably at the center of the gripper or at the tip of any tool. Set

$$y_n$$

= s in the direction of the gripper closure.

$$7.$$

Create a table of link parameters

$$a_i, \alpha_i , d_i, \theta_i$$

$$8.$$

From the homogeneous transformation matrices

$$A_i$$

by substituting the above parameters into

$$ \begin{aligned} A &= Rot_{z, \theta_i} Trans_{z, d_i} Trans_{x, a_i} Rot_{x, \alpha_i} \\ &= \begin{bmatrix} c_{\theta_i} & -s_{\theta_i}c_{\alpha_i} & s_{\theta_i}s_{\alpha_i} & a_i c_{\theta_i}\\ s_{\theta_i} & c_{\theta_i}c_{\alpha_i} & -c_{\theta_i}s_{\alpha_i} & a_i s_{\theta_i}\\ 0 & s_{\alpha_i} & c_{\alpha_i}& d_i\\ 0 & 0 & 0 & 1\\ \end{bmatrix} \end{aligned} $$ $$9.$$

Form

$$T^0_n = A_1 \cdots A_n$$

. This then gives the position and orientation of the tool frame expressed in base coordinates.