# 二輪ロボットに適したモデルは何ですか？

30

さまざまな忠実度のモデルを歓迎します。これには、非線形モデルと線形化モデルが含まれます。

1
この質問は非常に広いようです。「運動方程式」を、それが何であるかを説明するウィキペディアの記事（たとえば）にリンクすると役立ちます。また、ロボットをより具体的に指定する必要があります。たとえば、パッシブホイールはありますか？2つのホイールのタイプは何ですか？など
シャーバズ

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ポール

23

ここには多くの情報はありません。距離によって分離されてみましょうのは車輪を固定し、及び各車輪は配向性を有する、それらを結ぶ線に対して。次に、各車輪が角速度で独立して駆動できると仮定します。$b$$b$${\theta }_{i}$$\theta_i$${v}_{i}$$v_i$

これらの事のための通常のモデル、ベース速度であり、ベースの角速度では、次のとおりです。${v}_{b}$$v_b$${\omega }_{b}$$\omega_b$

${v}_{b}=\frac{1}{2}\cdot \left({v}_{1}+{v}_{2}\right)$
${\omega }_{b}=\frac{1}{b}\left({v}_{2}-{v}_{1}\right)$

そうロボット向きで開始仮定、および位置の時間窓に沿って、及び移動をと速度（左車輪）と（右輪）、それの向き意志である：$0$$0$$\left(0,0\right)$$(0,0)$$\delta t$$\delta t$${v}_{1}$$v_1$${v}_{2}$$v_2$との位置：

${\theta }_{1}=\frac{\delta t}{b}\left({v}_{2}-{v}_{1}\right)$
${p}_{x}=\mathrm{cos}\left(\frac{{\theta }_{1}}{2}\right)\cdot \left(2R\mathrm{sin}\left(\frac{{\theta }_{1}}{2}\right)\right)$
${p}_{y}=\mathrm{sin}\left(\frac{{\theta }_{1}}{2}\right)\cdot \left(2R\mathrm{sin}\left(\frac{{\theta }_{1}}{2}\right)\right)$

そのノート限界である ${v}_{1}\to {v}_{2}=v$$v_1\to v_2=v$

${p}_{x}=\delta t\cdot v$
${p}_{y}=0$

${p}_{x}$$p_x$

${p}_{x}=cos\left(\frac{{v}_{2}-{v}_{1}}{2b}\right)\ast 2\ast \left(b\frac{{v}_{1}+{v}_{2}}{2\left({v}_{2}-v1\right)}\right)\ast sin\left(\frac{{v}_{2}-{v}_{1}}{2b}\right)$

${p}_{x}=cos\left(\frac{{v}_{2}-{v}_{1}}{2b}\right)\ast \frac{\left({v}_{2}+{v}_{1}\right)}{2}\ast \frac{sin\left(\frac{{v}_{2}-{v}_{1}}{2b}\right)}{\frac{{v}_{2}-{v}_{1}}{2b}}$

Now note that we have three limits as ${v}_{2}\to {v}_{1}$$v_2 \rightarrow v_1$.

$cos\left(\frac{{v}_{2}-{v}_{1}}{2b}\right)\to 1$

$\frac{\left({v}_{2}+{v}_{1}\right)}{2}\to {v}_{1}=={v}_{2}$

This is covered all over the internet, but you might start here: http://rossum.sourceforge.net/papers/DiffSteer/ or here: https://web.cecs.pdx.edu/~mperkows/CLASS_479/S2006/kinematics-mobot.pdf

If the wheels are not fixed in direction, as in you can vary the speed and orientation, it gets more complicated. In that sense, a robot can become essentially holonomic (it can move in arbitrary directions and orientations on the plane). However, I bet for fixed orientation, you end up with the same model.

There are other models for two wheels, such as a bicycle model, which is easy to imagine as setting the velocities, and only varying one orientation.

That's the best I can do for now.

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Maybe I am a bit late but can not see why Px=dt*v if v1 = v2. We have sin(theta/2) as a part of multiplication therefore, when v1=v2 -> theta = 0, we get sin(0/2)=0 and as a consequence Px = 0. What I am missing?
Long Smith

In practice, just use the equations if $\theta \ne 0$$\theta \neq 0$. To answer your question, I've updated the answer.
Josh Vander Hook

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If you really want to dive into the mathematics of it, here's the seminal paper that unified and categorized most models for wheeled robots.

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I'm sorry, link-only answers are discouraged on StackExchange. Could you perhaps condense the content of that link into a few paragraphs and keep it here (along with the actual link, of course). This helps prevent link rot.
Manishearth

Sure thing, I'll do that as soon as I have enough time for it this week. Sorry about that, I wasn't aware about this policy and thought the link would be useful as is.
georgebrindeiro

Excellent paper - thanks for the link! Quite a long weekend as well :-)
uhoh

0

The answer to this is simple, but the other answers obfuscate the dynamics.

Differential drive robots can be modeled with unicycle dynamics of the form:

$\left[\begin{array}{c}\stackrel{˙}{x}\\ \stackrel{˙}{y}\\ \stackrel{˙}{\theta }\end{array}\right]=\left[\begin{array}{cc}cos\left(\theta \right)& 0\\ sin\left(\theta \right)& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}v\\ \omega \end{array}\right],$
where $x$$x$ and $y$$y$ are Cartesian coordinates of the robot, and $\theta \in \left(-\pi ,\pi \right]$$\theta \in (-\pi,\pi]$ is the angle between the heading and the $x$$x$-axis. The input vector ${\left[v,\omega \right]}^{T}$$\left[v, \omega \right]^T$ consists of linear and angular velocity inputs.

-1 This is merely a transformation between different coordinates. It doesn't model the dynamics of the robot at all as requested in the question. The "obfuscation" of the other answers is because they take into account that there are two wheels to control and not some abstract input vector. Such a vector could be the result of a model as requested in the question.
Bending Unit 22

The model that I have presented addresses the prompt, adds to the discussion, and is, in fact, a model of the dynamics of a non-holonomic differential drive robot (though not necessarily two-wheeled, which is a strength). While the input velocity vector (aka twist) may be an abstraction, using the twist input is standard for many two-wheeled platforms. This does, however, highlight the fact that state space representations are arbitrary. Controlling wheel velocities is an abstraction from controlling wheel torques, which is itself an abstraction from controlling motor currents.
JSycamore