# Product of exponentials formula for robot

From WiKi: Product of exponentials formula is:

$$g_{st}=e^{\hat{\xi}_1\theta_1}e^{\hat{\xi}_2\theta_2}...e^{\hat{\xi}_n\theta_n}$$

where $$g_{st}\in SE(3)$$,and $$\xi_i$$ is twist vector $$\in {se}(3)$$.

My question is: for programming convenient, can i use

$$g_{st}=e^{\hat{\xi}_1\theta_1+\hat{\xi}_2\theta_2+...+\hat{\xi}_n\theta_n}$$

to avoid the matrix product. If cannot, why?

I think the answer is yes, because $$\hat{\xi}_1\theta_1+...+\hat{\xi}_n\theta_n$$ is still in Lie algebra $$se(3)$$. But what is its physical explanation?

• No, there are Lie brackets terms coming from $\xi_i$ does not commute with $\xi_j$. See, for example, Baker-Campbell-Hausdorff formula – user10354138 Nov 13 '18 at 5:44
• @user10354138 I see, you are right, thank you. – Ben Nov 14 '18 at 8:32

The answer is NO, follows user10354138 comment, $$\xi_j$$ doesn't commute.