Introduction to Robotics II. Forward kinematics, MDH method

Preface

This chapter studies the forward kinematics of the manipulator ,MDH Law .

Joints and links

The joints joint, connecting rod link, It is the basic structure of the manipulator .

Joints include rotating joints and moving joints , Generally, there is only one degree of freedom .

One joint connects two adjacent connecting rods ,n Joint handle n+1 Connecting rods , have n A degree of freedom .

Number

Take the fixed base as the first 0 A connecting rod , The connecting rod at the end of the manipulator is used as the n A connecting rod .

Connecting rod parameters

Description of connecting rod

The length of the common vertical line between two adjacent joint axes , It is called the length of the connecting rod .

The angle formed between two adjacent joint axes , It is called the angle of the connecting rod .

Above picture , connecting rod i-1 The length of is its proximal joint axis i-1 With the distal joint axis i The length of the common vertical line between $a_{i-1}$

connecting rod i-1 The corner of is its proximal joint axis i-1 With the distal joint axis i Form an angle ${\alpha }_{i-1}$, As for the sign of angle , It can be determined when the coordinate system is established later .

Description of connecting rod connection

The distance between two adjacent connecting rods , It is called connecting rod offset .

The included angle of rotation of two adjacent connecting rods around the common joint axis , It is called joint angle .

Above picture , The joints i It's the connecting rod i-1 And connecting rod i The common joint of .

Because the actual connecting rod is bent , The common vertical line segment $a_{i-1}$ and $a_i$ As a straight connecting rod instead of a curved connecting rod i-1,i.

Straight link i-1,i The distance between them is the joint i Connecting rod offset $d_i$

Straight link i-1 Along the joint axis i Rotate to the straight link i The angle of is the joint angle ${\theta }_i$

Joint variables

A connecting rod can use the upper connecting rod length 、 Connecting rod angle 、 Connecting rod offset 、 Four parameters of joint angle are determined .

For rotating joints , The joint angle is variable , The other three parameters remain unchanged .

For moving joints , The connecting rod offset is variable , The other three parameters remain unchanged .

Variable parameters are called joint variables .

The connecting rod coordinate system

The connecting rod coordinate system is used to describe the relative position relationship between adjacent connecting rods .

The number of the connecting rod coordinate system is the same as that of the connecting rod , be called $\{i\}$.

The coordinate system of the intermediate connecting rod is established

For connecting rod i Set up a coordinate system ：

With joint axis i As Z Axis , With connecting rod i And Z Axis As origin , With connecting rod i As X Axis Point to the joint axis i+1, The right-hand rule determines Y Axis .

exception ： If the connecting rod i The length of $a_i=0$（ At this time, the connecting rod i,i+1 Of Z Axis The intersection ）, Take the intersection as origin , With Two Z The plane of the axis The vertical line of X Axis , There are two choices of direction , and ${\alpha }_i$ The symbol of is composed of X Axis Direction determines .

The establishment of each coordinate axis should meet the right-hand rule .

Above picture , The coordinate system can be established in this order ：

First find all the joint axes i-1,i

Then determine the coordinate system $\{i-1\}$, Take the joint axis as ${\hat{Z}}_{i-1}$, Straight link i-1 As ${\hat{X}}_{i-1}$, Then the right hand rule determines ${\hat{Y}}_{i-1}$.

Then determine the coordinate system $\{i\}$.

Establish the coordinate system of the head and tail connecting rods

For the head and tail connecting rods 0,n, There is a special way to build a department .

Head coordinate system

Coordinate system $\{0\}$ On the base , Generally used as a reference system .

The first joint variable is 0 when , Specify the coordinate system $\{0\}$ On $\{1\}$ coincidence .

When the first joint is a rotating joint ,$d_1=0$; The first joint is the mobile joint ,${\theta }_1=0$

Tail coordinate system

Coordinate system $\{n\}$ Origin and x Axis The direction can be arbitrarily selected , But try to make the connecting rod parameters 0.

Connecting rod parameters and connecting rod coordinate system

Follow the above method of building a department , The connecting rod parameters can be redefined ：

• $a_i$ Connecting rod length ： Along the ${\hat{X}}_i$, from ${\hat{Z}}_i$ Move to ${\hat{Z}}_{i+1}$ Distance of
• ${\alpha }_i$ Connecting rod torsion angle ： Around the ${\hat{X}}_i$ Axis , hold ${\hat{Z}}_i$ Rotate to ${\hat{Z}}_{i+1}$ The angle of
• $d_i$ Connecting rod offset ： Along the ${\hat{Z}}_i$, from ${\hat{X}}_{i-1}$ Move to ${\hat{X}}_{i1}$ Distance of
• ${\theta }_i$ Joint angle ： Around the ${\hat{Z}}_i$ Axis , hold ${\hat{X}}_{i-1}$ Rotate to ${\hat{X}}_i$ The angle of

Set up $a_i>0$, Other parameters can be positive or negative .

The above method of establishing the connecting rod coordinate system and connecting rod parameters is called MDH Law (Modified Denavit–Hartenberg).

DH The coordinate system established by the method is not unique .

Connecting rod change

Connecting rod parameters can be used to calculate the relative pose between adjacent rods .

Above picture , Consider the coordinate system $\{i-1\},\{i\}$ The transformation between ${}_{i-1}^{i}T$, Establish an intermediate coordinate system $\{P\},\{Q\},\{R\}$, Then there are ：
$${}_i^{i-1}T={}_R^{i-1}T{}_Q^{R}T{}_P^{Q}T{}_i^{P}T$$
The transformation above , It can be seen as putting ${\hat{X}}_i$ Transformation for ${\hat{X}}_{i-1}$, Then there are ：
$${}_i^{i-1}T=R_X({\alpha }_{i-1})D_X(a_{i-1})R_Z({\theta }_i)D_Z(d_i)$$
Or write out the transformation of each intermediate coordinate system ：
$$\begin{gathered} {}_i^{P}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& d_i\\ 0& 0& 0& 1\end{array}\right] \\ {}_P^{Q}T=\left[\begin{array}{cccc}\cos {\theta }_1& -\sin {\theta }_1& 0& 0\\ \sin {\theta }_1& \cos {\theta }_1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right] \\ {}_Q^{R}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& a_{i-1}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right] \\ {}_R^{i-1}T=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\alpha }_{i-1}& -\sin {\alpha }_{i-1}& 0\\ 0& \sin {\alpha }_{i-1}& \cos {\alpha }_{i-1}& 0\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$
obtain ${}_i^{i-1}T$ The general form of ：

MDH Method use steps

1. Find all joint axes ;
2. Establish the coordinate system of the intermediate connecting rod in sequence ;
3. Determine the head and tail coordinate system ;
4. Write DH Parameters ;
5. Calculate the forward kinematics matrix .

Postscript

This article is about forward kinematics in robotics , It's using MDH Law .

There will be Matlab Example programming .