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DH parameters of robotics and derivation using MATLAB symbolic operation

2022-06-28 04:55:00 【subtitle_】

introduction
Reviewed robotics again DH Parameters , And make use of matlab Symbolic operation is derived , The formula is verified .

introduction
- 1.DH Parameter principle
- 2.Matlab Symbolic operation verification

1.DH Parameter principle

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In the picture Coordinate system definition ：

Coordinate system ${i}$ Of $z$ Axis $z_i$ And joint axis $i$ Collinear , Point to any regulation .
Coordinate system ${i}$ Of $x$ Axis $x_i$ and $a_i$ coincidence , By joint $i$ Point to the joint $i + 1$ , When $a_i=0$ , take $x_i=±z_{i+1}\times z_i$ .
Coordinate system ${i}$ Of $y$ Axis $y_i$ According to the right-hand rule .
Coordinate system ${i}$ The origin of $o_i$ Take from $x_i$ and $z_i$ At the intersection of ; When $z_i$ and $z_{i+}$ When we meet , The origin is at the intersection of the two axes , When $z_i$ and $z_{i+}$ Parallel time , Take the origin to make $d_{i+1}=0$ The place of .

Use the connecting rod coordinate system to define the corresponding Connecting rod parameters ：

$a_i$ = from $z_i$ To $z_{i+1}$ Along the $x_i$ Measured distance （ Length of common vertical line ）
$\alpha_i$ = from $z_i$ To $z_{i+1}$ Along the $x_i$ The angle of rotation
$d_i$ = from $x_{i-1}$ To $x_{i}$ Along the $z_i$ Measured distance
$\theta_i$ = from $x_{i-1}$ To $x_{i}$ Along the $z_i$ The angle of rotation

Coordinate system ${i\}$ Relative coordinates ${i-1\}$ The transformation matrix of ${}^{i-1}T_{i}$ It can be regarded as the product of four sub transformation matrices ：

Around the $x_{i-1}$ pivot $\alpha_{i-1}$ horn
$t_{x}\left(\alpha_{i-1}\right)=\begin{bmatrix}1 &0&0&0\\ 0&cos\alpha_{i-1}&{-sin\alpha_{i-1}}&0\\0&{sin\alpha_{i-1}}&{cos\alpha_{i-1}}&{0}\\ {0}&{0}&{0}&{1}\end{bmatrix}$
Along the $x_{i-1}$ Axis movement $a_{i-1}$
$s_{x}\left(a_{i-1}\right)=\begin{bmatrix}1 &0&0&a_{i-1}\\ 0&1&0&0\\0&0&1&{0}\\ {0}&{0}&{0}&{1}\end{bmatrix}$
Along the $z_{i}$ Axis movement $\theta_{i}$
$t_{z}\left(\theta_{i}\right)=\begin{bmatrix}{cos\theta_{i}}&{-sin\theta_{i}}&0&0\\ {sin\theta_{i}}&{cos\theta_{i}}&0&0\\0&0&1&0\\ {0}&{0}&{0}&{1}\end{bmatrix}$
Along the $z_{i}$ Axis movement $d_{i}$
$s_{z}\left(d_{i}\right)=\begin{bmatrix}1 &0&0&0\\ 0&1&0&0\\0&0&1&d_i\\ {0}&{0}&{0}&{1}\end{bmatrix}$

These transformations are described in the relative motion coordinate system , Satisfy the principle from left to right , The final transformation formula can be obtained as follows ：

$_{i}^{i-1}T=R o t_{x}\left(\alpha_{i}\right)T r a n s_{x}\left(a_{i}\right)R o t_{z}\left(\theta_{i}\right)T r a n s_{z}\left(d_{i}\right)$
It is usually stipulated that $a_i\ge0$ , Because it represents the length of the connecting rod , and $\alpha_i, d_i, \theta_i$ Positive but negative .

$_{i}^{i-1}T=\begin{bmatrix}{cos\theta_{i}}&{-sin\theta_{i}}&0&{a _{i-1}}\\ {sin\theta_{i}cos\alpha_{i-1}}&{cos\theta_{i}cos\alpha_{i-1}}&{-sin\alpha_{i-1}}&{-d_isin\alpha_{i-1}}\\{sin\theta_{i}sin\alpha_{i-1}}&{cos\theta_{i}sin\alpha_{i-1}}&{cos\alpha_{i-1}}&{d_icos\alpha_{i-1}}\\ {0}&{0}&{0}&{1}\end{bmatrix}$

2.Matlab Symbolic operation verification

utilize Matlab The symbolic operation of is used to deduce the formula and verify the formula

syms theta d alpha a
M=DH_Trans(theta,d,alpha,a)
function M=DH_Trans(theta,d,alpha,a)
    R_z_theta=[cos(theta),-sin(theta),0,0;...
        sin(theta),cos(theta),0,0;...
        0,0,1,0;...
        0,0,0,1];
    T_z_d=[1,0,0,0;...
        0,1,0,0;...
        0,0,1,d;...
        0,0,0,1];
    R_x_alpha=[1,0,0,0;...
        0,cos(alpha),-sin(alpha),0;...
        0,sin(alpha),cos(alpha),0;...
        0,0,0,1];
   T_x_a=[1,0,0,a;...
        0,1,0,0;...
        0,0,1,0;...
        0,0,0,1];
   M=R_x_alpha*T_x_a*R_z_theta*T_z_d;
end

The results are consistent
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