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为 3D 无人机添加角度

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为 3D 无人机添加角度

我怎样才能像图片一样将欧拉角和 XYZ 角添加到代码中？在此处输入图片描述

\documentclass[tikz]{standalone}
%\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{3d, arrows.meta, calc, quotes}
\begin{document}
\begin{tikzpicture}[
  x = ( 20:-2cm), y = (-30: 2cm), z = ( 80: 1.6cm),
  > = Triangle]
\draw coordinate (O) (0,0,-.05) ellipse [radius=.2];

\foreach \xy/\yx in {x/y, y/x} {
  \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
            -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
  \foreach \lmr in {-.05, 0, .05}
    \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
  \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                (xyz cs: \xy =  1) ellipse[];
}
\draw[fill=white] (O) ellipse [radius=.2];

\foreach \xyz/\Label/\Ang in {% \Ang needs to start with a ,
  x/left/ {, longitude = 70},
  y/right/{, longitude =-20},
  z/above/{, longitude = 70, latitude=70}%
} {
  \draw[->, densely dashed, red] (0, 0, 0) -- ++(xyz cs: \xyz= 1.5)
    coordinate (ax-\xyz) node[\Label] {$\xyz$};
  \draw[blue, ->]
     (0, 0, 0) -- (xyz spherical cs: radius = 1.5,/tikz/style/.expanded=\Ang)
      coordinate (ax-\xyz') node[\Label] {$\xyz$};
}

\draw[canvas is xy plane at z=0]
  (0, 1) arc[start angle=90, delta angle=20, radius=1]
  node[midway, below right] {$\varphi$};
\end{tikzpicture}
\end{document}

答案1

我建议使用该tikz-3dplot包。包中已经有一个使用欧拉角定义的坐标变换。您可以找到类似的问题，其中包含一些有用的答案这里。基本上，您可以定义一个主坐标系，您可以使用两个角度设置其方向（例如\tdplotsetmaincoords{50}{115}）。然后，可以通过命令旋转此坐标系\tdplotsetrotatedcoords{<angle1>}{<angle2>}{<angle3>}。在包中，此命令用于使用 zyz 序列中的欧拉角执行坐标系旋转。

为了得到另一个序列，可以重新定义转换命令，如在这和这个答案到之前提到的问题。这样，您最终可以添加所有可能的序列并在它们之间切换。在下面的代码中，我使用了 zyx 序列（称为 Tait-Bryan 或 Cardan 角（参见维基百科中的欧拉角))，因为它最接近给定的图片，也是 Tait-Byran 角中最常见的，并且也是 zxz 序列，作为最常见的经典欧拉角。

然后，您可以在主坐标系或旋转坐标系中绘制。这样，就可以轻松添加任何 3D 绘图，例如您的无人机以及用于标记角度的弧线。该tikz-3dplot软件包包含一个\tdplotdrawarc[coordinates_frame,->,colo]{origin}{radius}{start angle}{end angle}{node info}{label}命令，但我发现使用您已经使用的选项在旋转坐标系的平面上手动绘制弧线更直观canvas is xy plane at z=0。

\documentclass[tikz]{standalone}
%\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, quotes}
%\usetikzlibrary{3d, arrows.meta, calc, quotes}

\usepackage{tikz-3dplot}

% Define new arc command
% Syntax: [draw options] (center) (initial angle:final angle:radius)
% https://tex.stackexchange.com/a/66220
\def\centerarc[#1](#2)(#3:#4:#5){%
    \draw [#1] ($(#2)+({#5*cos(#3)}, {#5*sin(#3)})$) arc (#3:#4:#5)%
}
 
% Redefine the rotation sequence for the tikz3d-plot to Euler-Angles:
% z-x-z with alpha-beta-gamma (psi-theta-phi)
% https://tex.stackexchange.com/q/118069/98906
\newcommand{\tdseteulerzxz}{%
    \renewcommand{\tdplotcalctransformrotmain}{%
        %
        % Determine the sin and cos of the specified angle in degrees
        % \tdplotsinandcos{sin}{cos}{theta}
        % - #1: Returns sin(#3)
        % - #2: Returns cos(#3)
        % - #3: User-specified angle theta
        \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
        \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
        \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
        %
        % Define trigonometric abbreviations
        %
        \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
        \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
        \tdplotmult{\sacbsg}{\sacb}{\singamma}
        \tdplotmult{\sacbcg}{\sacb}{\cosgamma}
        \tdplotmult{\sasg}{\sinalpha}{\singamma}
        \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
        %
        \tdplotmult{\sbsg}{\sinbeta}{\singamma}
        \tdplotmult{\sbcg}{\sinbeta}{\cosgamma}
        %
        \tdplotmult{\casb}{\cosalpha}{\sinbeta}
        \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
        \tdplotmult{\cacbsg}{\cacb}{\singamma}
        \tdplotmult{\cacbcg}{\cacb}{\cosgamma}
        \tdplotmult{\casg}{\cosalpha}{\singamma}
        \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
        %
        % Define the entries for the rotation matrix from the B-System to the I-System
        % This is A_IB = (A_BI)^T
        %
        \pgfmathsetmacro{\raaeul}{+\cacg - \sacbsg}
        \pgfmathsetmacro{\rabeul}{-\casg - \sacbcg}
        \pgfmathsetmacro{\raceul}{+\sasb}
        %
        \pgfmathsetmacro{\rbaeul}{+\sacg + \cacbsg}
        \pgfmathsetmacro{\rbbeul}{-\sasg + \cacbcg}
        \pgfmathsetmacro{\rbceul}{-\casb}
        %
        \pgfmathsetmacro{\rcaeul}{+\sbsg}
        \pgfmathsetmacro{\rcbeul}{+\sbcg}
        \pgfmathsetmacro{\rcceul}{+\cosbeta}
        %
    }
}
 
% Redefine the rotation sequence for the tikz3d-plot to Cardan-Angles:
% z-y-x with alpha-beta-gamma
% https://tex.stackexchange.com/q/118069/98906
\newcommand{\tdsetcardanzyx}{%
    \renewcommand{\tdplotcalctransformrotmain}{%
        %
        % Determine the sin and cos of the specified angle in degrees
        % \tdplotsinandcos{sin}{cos}{theta}
        % - #1: Returns sin(#3)
        % - #2: Returns cos(#3)
        % - #3: User-specified angle theta
        \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
        \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
        \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
        %
        % Define trigonometric abbreviations
        %
        \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
        \tdplotmult{\sasbsg}{\sasb}{\singamma}
        \tdplotmult{\sasbcg}{\sasb}{\cosgamma}
        %
        \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
        \tdplotmult{\sasg}{\sinalpha}{\singamma}
        \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
        %
        \tdplotmult{\casb}{\cosalpha}{\sinbeta}
        \tdplotmult{\casbsg}{\casb}{\singamma}
        \tdplotmult{\casbcg}{\casb}{\cosgamma}
        %
        \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
        \tdplotmult{\casg}{\cosalpha}{\singamma}
        \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
        \tdplotmult{\cbsg}{\cosbeta}{\singamma}
        \tdplotmult{\cbcg}{\cosbeta}{\cosgamma}
        %
        % Define the entries for the rotation matrix from the B-System to the I-System
        % This is A_IB = (A_BI)^T
        %
        \pgfmathsetmacro{\raaeul}{+\cacb}
        \pgfmathsetmacro{\rabeul}{+\casbsg - \sacg}
        \pgfmathsetmacro{\raceul}{+\sasg+\casbcg}
        %
        \pgfmathsetmacro{\rbaeul}{+\sacb}
        \pgfmathsetmacro{\rbbeul}{+\cacg + \sasbsg}
        \pgfmathsetmacro{\rbceul}{\sasbcg-\casg}
        %
        \pgfmathsetmacro{\rcaeul}{-\sinbeta}
        \pgfmathsetmacro{\rcbeul}{+\cbsg}
        \pgfmathsetmacro{\rcceul}{+\cbcg}
        %
    }
}
 
% Plot display orientation
\tdplotsetmaincoords{50}{115}

\begin{document}


% z-y-x Tait-Bryan angles
\begin{tikzpicture}[scale=2,tdplot_main_coords,> = Triangle]
    % change ratation to z-y-x Tait-Bryan / Cardan angles
    \tdsetcardanzyx
    
    \pgfmathsetmacro{\angpsi}{-20}  % or alpha
    \pgfmathsetmacro{\angtheta}{-10}    % or beta
    \pgfmathsetmacro{\angphi}{-5}   % or gamma

    \pgfmathsetmacro{\r}{2} % unit length radius      
    
    \coordinate (O) at (0,0,0);     
    
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords,scale=1.2]
        \draw coordinate (O) (0,0,-.05) ellipse [radius=.2];
        \foreach \xy/\yx in {x/y, y/x} {
          \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
                    -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
          \foreach \lmr in {-.05, 0, .05}
            \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
          \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                        (xyz cs: \xy =  1) ellipse[];
        }
        \draw[fill=white] (0,0,0) ellipse [radius=.2];
        
    \end{scope}     
    
    
    % original coordinate system
    \foreach \xyz/\Label in {x/left,y/right,z/above}
        \draw[->,blue] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz$};
    
    % angle of first rotation around z-axis
    \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angpsi:1.5);
    \node at ({1.7*cos(\angpsi/2)},{1.7*sin(\angpsi/2)},0){$\psi$};
    \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angpsi:1.5);
    \node at ({-1.7*sin(\angpsi/2)},{1.7*cos(\angpsi/2)},0){$\psi$};    
    \tdplotsetrotatedcoords{\angpsi}{0}{0}
    
    % new coordinate system
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right}
            \draw[->,green] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz'$};
        
        % angle of second rotation around new y-axis
        \draw[canvas is xz plane at y=0,->] (1.5,0) arc (0:-\angtheta:1.5);
        \node at ({1.7*cos(\angtheta/2)},0,{-1.7*sin(\angtheta/2)}){$\theta$};
        \draw[canvas is xz plane at y=0,->] (0,1.5) arc (90:90-\angtheta:1.5);
        \node at ({1.7*sin(\angtheta/2)},0,{1.7*cos(\angtheta/2)}){$\theta$};
    \end{scope} 
    
        
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{0}  
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {z/above}
            \draw[->,magenta] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz''$};
        
        % angle of third rotation around new x-axis
        \draw[canvas is yz plane at x=0,->] (1.5,0) arc (0:\angphi:1.5);
        \node at (0,{1.7*cos(\angphi/2)},{1.7*sin(\angphi/2)}){$\varphi$};
        \draw[canvas is yz plane at x=0,->] (0,1.5) arc (90:90+\angphi:1.5);
        \node at (0,{-1.7*sin(\angphi/2)},{1.7*cos(\angphi/2)}){$\varphi$};
    \end{scope}     
    
    % rotated coordinate system
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right,z/above}
            \draw[->,densely dashed,red] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\tilde\xyz$};
    \end{scope}
\end{tikzpicture}

% z-x-z Euler angles
\begin{tikzpicture}[scale=2,tdplot_main_coords,> = Triangle]
    % change ratation to z-y-x Tait-Bryan / Cardan angles
    \tdseteulerzxz
    
    \pgfmathsetmacro{\angpsi}{-10}  % or alpha
    \pgfmathsetmacro{\angtheta}{-20}    % or beta
    \pgfmathsetmacro{\angphi}{-10}  % or gamma

    \pgfmathsetmacro{\r}{2} % unit length radius      
    
    \coordinate (O) at (0,0,0);     
    
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords,scale=1.2]
        \draw coordinate (O) (0,0,-.05) ellipse [radius=.2];
        \foreach \xy/\yx in {x/y, y/x} {
          \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
                    -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
          \foreach \lmr in {-.05, 0, .05}
            \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
          \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                        (xyz cs: \xy =  1) ellipse[];
        }
        \draw[fill=white] (0,0,0) ellipse [radius=.2];
        
    \end{scope}     
    
    
    % original coordinate system
    \foreach \xyz/\Label in {x/left,y/right,z/above}
        \draw[->,blue] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz$};
    
    % angle of first rotation around z-axis
    \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angpsi:1.5);
    \node at ({1.7*cos(\angpsi/2)},{1.7*sin(\angpsi/2)},0){$\psi$};
    \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angpsi:1.5);
    \node at ({-1.7*sin(\angpsi/2)},{1.7*cos(\angpsi/2)},0){$\psi$};    
    \tdplotsetrotatedcoords{\angpsi}{0}{0}
    
    % new coordinate system
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right}
            \draw[->,green] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz'$};
        
        % angle of second rotation around new x-axis
        \draw[canvas is yz plane at x=0,->] (1.5,0) arc (0:\angtheta:1.5);
        \node at (0,{1.7*cos(\angtheta/2)},{1.7*sin(\angtheta/2)}){$\theta$};
        \draw[canvas is yz plane at x=0,->] (0,1.5) arc (90:90+\angtheta:1.5);
        \node at (0,{-1.7*sin(\angtheta/2)},{1.7*cos(\angtheta/2)}){$\theta$};
    \end{scope} 
    
        
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{0}  
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {y/right}
            \draw[->,magenta] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz''$};
        
        % angle of third rotation around new z-axis
        \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angphi:1.5);
        \node at ({1.7*cos(\angphi/2)},{1.7*sin(\angphi/2)},0){$\varphi$};
        \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angphi:1.5);
        \node at ({-1.7*sin(\angphi/2)},{1.7*cos(\angphi/2)},0){$\varphi$};
    \end{scope}     
    
    % rotated coordinate system
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right,z/above}
            \draw[->,densely dashed,red] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\tilde\xyz$};
    \end{scope}
\end{tikzpicture}


\end{document}

结果：

Tait-Bryann zyx Tait-Bryan_zyx 欧拉 欧拉

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