SDH(标准DH)和MDH(改进DH)
改进DH坐标系建立如图1所示，标准DH坐标系建立如图2所示。改进DH和标准DH的主要区别为：
- 连杆坐标系建立的位置不同。SDH方法将连杆的坐标系固定在连杆的后端，MDH方法将连杆的坐标系固定在连杆的前端
- 变换的顺序不同。SDH方法的变换顺序为 d d d→ θ \theta θ→ a a a→ α \alpha α，MDH方法的变换顺序为 α \alpha α→ a a a→ θ \theta θ→ d d d。
SDH方法的变换矩阵为：
i − 1 i T = R o t z i − 1 ( θ i ) T r a n s z i − 1 ( d i ) T r a n s x i ( a i ) R o t x i ( α i ) = [ c o s θ i − s i n θ i 0 0 s i n θ i c o s θ i 0 0 0 0 1 0 0 0 0 1 ] ⋅ [ 1 0 0 0 0 1 0 0 0 0 0 d i 0 0 1 1 ] ⋅ [ 1 0 0 a i 0 1 0 0 0 0 1 0 0 0 0 1 ] ⋅ [ 1 0 0 0 0 c o s α i − s i n α i 0 0 s i n α i c o s α i 0 0 0 0 1 ] = [ 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 ] _{i-1}^{i}\textrm{T}=Rot_{z_{i-1}}(\theta_i)Trans_{z_{i-1}}(d_i)Trans_{x_i}(a_i)Rot_{x_i}(\alpha_i)\\ =\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}\cdot \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & d_i\\ 0 & 0 & 1 & 1 \end{bmatrix}\cdot \begin{bmatrix} 1 & 0 & 0 & a_i\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & cos\alpha_i & -sin\alpha_i & 0\\ 0 & sin\alpha_i & cos\alpha_i & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}\\ =\begin{bmatrix} c\theta_i & -s\theta_i \cdot c\alpha_i & s\theta_i \cdot s\alpha_i & a_i c\theta_i\\ s\theta_i & c\theta_i \cdot c\alpha_i & -c\theta_i \cdot s\alpha_i & a_i s\theta_i\\ 0 & s\alpha_i & c\alpha_i & d_i\\ 0 & 0 & 0 & 1 \end{bmatrix} i−1iT=Rotzi−1(θi)Transzi−1(di)Transxi(ai)Rotxi(αi)=⎣ ⎡cosθisinθi00−sinθicosθi0000100001⎦ ⎤⋅⎣ ⎡10000100000100di1⎦ ⎤⋅⎣ ⎡100001000010ai001⎦ ⎤⋅⎣ ⎡10000cosαisinαi00−sinαicosαi00001⎦ ⎤=⎣ ⎡cθisθi00−sθi⋅cαicθi⋅cαisαi0sθi⋅sαi−cθi⋅sαicαi0aicθiaisθidi1⎦ ⎤
MDH方法的变换矩阵为：
i − 1 i T = R o t x i − 1 ( α i − 1 ) T r a n s x i − 1 ( a i − 1 ) R o t z i ( θ i ) T r a n s z i ( d i ) = [ 1 0 0 0 0 c o s α i − s i n α i 0 0 s i n α i c o s α i 0 0 0 0 1 ] ⋅ [ 1 0 0 a i 0 1 0 0 0 0 1 0 0 0 0 1 ] ⋅ [ c o s θ i − s i n θ i 0 0 s i n θ i c o s θ i 0 0 0 0 1 0 0 0 0 1 ] ⋅ [ 1 0 0 0 0 1 0 0 0 0 0 d i 0 0 1 1 ] = [ c θ i − s θ i 0 a i − 1 s θ i c α i − 1 c θ i c α i − 1 − s α i − 1 − d i s α i − 1 s θ i s α i − 1 c θ i s α i − 1 c α i − 1 d i c α i − 1 0 0 0 1 ] _{i-1}^{i}\textrm{T}=Rot_{x_{i-1}}(\alpha_{i-1})Trans_{x_{i-1}}(a_{i-1})Rot_{z_i}(\theta_i)Trans_{z_i}(d_i)\\ = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & cos\alpha_i & -sin\alpha_i & 0\\ 0 & sin\alpha_i & cos\alpha_i & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}\cdot \begin{bmatrix} 1 & 0 & 0 & a_i\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix} \cdot \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}\cdot \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & d_i\\ 0 & 0 & 1 & 1 \end{bmatrix} \\ =\begin{bmatrix} c\theta_i & -s\theta_i & 0 & a_{i-1} \\ s\theta_ic\alpha_{i-1} & c\theta_ic\alpha_{i-1} & -s\alpha_{i-1} & -d_is\alpha_{i-1} \\ s\theta_is\alpha_{i-1} & c\theta_is\alpha_{i-1} & c\alpha_{i-1} & d_ic\alpha_{i-1} \\ 0 & 0 & 0 & 1 \end{bmatrix} i−1iT=Rotxi−1(αi−1)Transxi−1(ai−1)Rotzi(θi)Transzi(di)=⎣ ⎡10000cosαisinαi00−sinαicosαi00001⎦ ⎤⋅⎣ ⎡100001000010ai001⎦ ⎤⋅⎣ ⎡cosθisinθi00−sinθicosθi0000100001⎦ ⎤⋅⎣ ⎡10000100000100di1⎦ ⎤=⎣ ⎡cθisθicαi−1sθisαi−10−sθicθicαi−1cθisαi−100−sαi−1cαi−10ai−1−disαi−1dicαi−11⎦ ⎤
| MDH | SDH |
|---|---|
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对于平面RRR机械臂，其MDH和SDH的DH参数坐标系建立如上面两图所示。MDH方法的坐标系 { 0 } \{ 0 \} {0}和坐标系 { 1 } \{ 1 \} {1}重合，坐标系 { 2 } \{ 2 \} {2}和坐标系 { 3 } \{ 3 \} {3}重合，坐标系建立在连杆前端。SDH方法的坐标系建立在连杆后端。因此MDH和SDH方法的DH参数表如下：
MDH方法DH参数表：
| i i i | α i − 1 \alpha_{i-1} αi−1 | a i − 1 a_{i-1} ai−1 | d i d_i di | θ i \theta_i θi |
|---|---|---|---|---|
| 1 | 0 | 0 | 0 | θ 1 \theta_1 θ1 |
| 2 | 0 | L 1 L_1 L1 | 0 | θ 2 \theta_2 θ2 |
| 3 | 0 | L 2 L_2 L2 | 0 | θ 3 \theta_3 θ3 |
SDH方法DH参数表：
| i i i | θ i \theta_i θi | d i d_i di | a i a_i ai | α i \alpha_i αi |
|---|---|---|---|---|
| 1 | 0 | L 1 L_1 L1 | 0 | θ 1 \theta_1 θ1 |
| 2 | 0 | L 2 L_2 L2 | 0 | θ 2 \theta_2 θ2 |
| 3 | 0 | L 3 L_3 L3 | 0 | θ 3 \theta_3 θ3 |