文章目錄

• 一、繪制scara機器人工作空間
• 二、MATLAB代碼

一、繪制scara機器人工作空間

??如上圖，scara機器人大臂長$L_1$，小臂長$L_2$，${\theta }_1$為關節1角度，${\theta }_2$為關節2角度，且${\theta }_1\in [{\theta }_{L1},{\theta }_{U1}],{\theta }_2\in [{\theta }_{L2},{\theta }_{U2}]$。

??scara機器人工作空間由四段圓弧組成(如上圖)，圓弧方程如下：
$$?\begin{array}{l}{x}^2+y^2=(L_1+L_2{)}^2\\ x^2+y^2=L_1^2+L_2^2+2L_1{L}_{2}cos\theta \\ (x?L_{1}cos{\theta }_{L1}{)}^2+(y?L_{1}sin{\theta }_{L1}{)}^2=L_2^2\\ (x?L_{1}cos{\theta }_{U1}{)}^2+(y?L_{1}sin{\theta }_{U1}{)}^2=L_2^2\end{array}\text{\hspace{1em}(1)}$$
??其中，左手系時，$\theta ={\theta }_{L2}$；右手系時，$\theta ={\theta }_{U2}$。

二、MATLAB代碼

??繪制2D圓弧：

%{ Function: draw_2d_arc Description: 繪制平面圓弧 Input: 圓弧圓心(x0, y0),半徑r，起始角度theta1(rad)，結束角度theta2(rad), 曲線樣式選擇options Output: 無 Author: Marc Pony(marc_pony@163.com) %} function draw_2d_arc(x0, y0, r, theta1, theta2, options) deltaTheta = 0.1 * pi / 180; theta = theta1 : deltaTheta : theta2; x = x0 + r * cos(theta); y = y0 + r * sin(theta); plot(x, y, 'LineStyle', options.LineStyle, 'Color', options.Color, 'LineWidth', options.LineWidth); axis equal; end

??繪制scara機器人工作空間：

%{ Function: draw_scara_workspace Description: 繪制scara機器人工作空間 Input: 大臂L1，小臂L2，關節1限位角度thetaLimit1(rad)，關節2限位角度thetaLimit2(rad)，手系handcoor Output: 無 Author: Marc Pony(marc_pony@163.com) %} function draw_scara_workspace(L1, L2, thetaLimit1, thetaLimit2, handcoor) thetaL1 = thetaLimit1(1); thetaU1 = thetaLimit1(2); thetaL2 = thetaLimit2(1); thetaU2 = thetaLimit2(2);hold on; if(handcoor == 1) %right handcooroptions.LineStyle = '-';options.Color='g';options.LineWidth = 3;x0 = 0;y0 = 0;r = sqrt(L1 * L1 + L2 * L2 + 2 * L1 * L2 * cos(thetaU2));alpha = acos((r^2 + L1^2 - L2^2) / (2 * r * L1));thetaStart = thetaL1 + alpha;thetaEnd = thetaU1 + alpha;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = 0;y0 = 0;r = L1 + L2;thetaStart = thetaL1;thetaEnd = thetaU1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = L1 * cos(thetaU1);y0 = L1 * sin(thetaU1);r = L2;thetaStart = thetaU1;thetaEnd = thetaU1 + thetaU2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);x0 = L1 * cos(thetaL1);y0 = L1 * sin(thetaL1);r = L2;thetaStart = thetaL1;thetaEnd = thetaL1 + thetaU2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);title('Workspace in right handcoor', 'fontsize', 16); else %left handcooroptions.LineStyle = '-';options.Color='b';options.LineWidth = 3;x0 = 0;y0 = 0;r = sqrt(L1 * L1 + L2 * L2 + 2 * L1 * L2 * cos(thetaL2));alpha = acos((r^2 + L1^2 - L2^2) / (2 * r * L1));thetaStart = thetaL1 - alpha;thetaEnd = thetaU1 - alpha;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = 0;y0 = 0;r = L1 + L2;thetaStart = thetaL1;thetaEnd = thetaU1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = L1 * cos(thetaU1);y0 = L1 * sin(thetaU1);r = L2;thetaStart = thetaU1 + thetaL2;thetaEnd = thetaU1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);x0 = L1 * cos(thetaL1);y0 = L1 * sin(thetaL1);r = L2;thetaStart = thetaL1 + thetaL2;thetaEnd = thetaL1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);title('Workspace in left handcoor', 'fontsize', 16); end set(gcf, 'color', 'w'); axis off; end

??繪制scara機器人工作空間草圖：

%{ Function: draw_scara_workspace_sketch Description: 繪制scara機器人工作空間草圖 Input: 大臂L1，小臂L2，關節1限位角度thetaLimit1(rad)，關節2限位角度thetaLimit2(rad)，手系handcoor Output: 無 Author: Marc Pony(marc_pony@163.com) %} function draw_scara_workspace_sketch(L1, L2, thetaLimit1, thetaLimit2, handcoor)thetaL1 = thetaLimit1(1); thetaU1 = thetaLimit1(2); thetaL2 = thetaLimit2(1); thetaU2 = thetaLimit2(2);hold on; if(handcoor == 1) %right handcooroptions.LineStyle = '-';options.Color='g';options.LineWidth = 3;x0 = 0;y0 = 0;r = sqrt(L1 * L1 + L2 * L2 + 2 * L1 * L2 * cos(thetaU2));alpha = acos((r^2 + L1^2 - L2^2) / (2 * r * L1));thetaStart = thetaL1 + alpha;thetaEnd = thetaU1 + alpha;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = 0;y0 = 0;r = L1 + L2;thetaStart = thetaL1;thetaEnd = thetaU1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = L1 * cos(thetaU1);y0 = L1 * sin(thetaU1);r = L2;thetaStart = thetaU1;thetaEnd = thetaU1 + thetaU2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);x0 = L1 * cos(thetaL1);y0 = L1 * sin(thetaL1);r = L2;thetaStart = thetaL1;thetaEnd = thetaL1 + thetaU2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);%-------------options.LineStyle = '--';options.Color='r';options.LineWidth = 0.5;x0 = 0;y0 = 0;r = sqrt(L1 * L1 + L2 * L2 + 2 * L1 * L2 * cos(thetaU2));thetaStart = 0;thetaEnd = 2 * pi;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)r = L1 + L2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = L1 * cos(thetaU1);y0 = L1 * sin(thetaU1);r = L2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);x0 = L1 * cos(thetaL1);y0 = L1 * sin(thetaL1);draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);xA1 = L1 * cos(thetaL1);yA1 = L1 * sin(thetaL1);xB1 = xA1 + L2 * cos(thetaL1 + thetaU2);yB1 = yA1 + L2 * sin(thetaL1 + thetaU2);xA2 = L1 * cos(thetaU1);yA2 = L1 * sin(thetaU1);xB2 = xA2 + L2 * cos(thetaU1 + thetaU2);yB2 = yA2 + L2 * sin(thetaU1 + thetaU2);xC1 = (L1 + L2) * cos(thetaL1);yC1 = (L1 + L2) * sin(thetaL1);xC2 = (L1 + L2) * cos(thetaU1);yC2 = (L1 + L2) * sin(thetaU1);plot([0, xA1, xB1], [0, yA1, yB1], 'lineStyle', '-', 'color', 'k', 'lineWidth', 3);plot([0, xA2, xB2], [0, yA2, yB2], 'lineStyle', ':', 'color', 'k', 'lineWidth', 3);fontsize = 15;delta = 25;text(0, 0, 'O', 'Fontsize', fontsize);text(xA1, yA1 - delta, 'A_1', 'fontsize', fontsize);text(xB1, yB1 - delta, 'B_1', 'fontsize', fontsize);text(xA2, yA2 + delta, 'A_2', 'fontsize', fontsize);text(xB2, yB2 - delta, 'B_2', 'fontsize', fontsize);text(xC1, yC1, 'C_1', 'fontsize', fontsize);text(xC2, yC2, 'C_2', 'fontsize', fontsize);title('Workspace sketch in right handcoor', 'fontsize', 16);else %left handcooroptions.LineStyle = '-';options.Color='b';options.LineWidth = 3;x0 = 0;y0 = 0;r = sqrt(L1 * L1 + L2 * L2 + 2 * L1 * L2 * cos(thetaL2));alpha = acos((r^2 + L1^2 - L2^2) / (2 * r * L1));thetaStart = thetaL1 - alpha;thetaEnd = thetaU1 - alpha;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = 0;y0 = 0;r = L1 + L2;thetaStart = thetaL1;thetaEnd = thetaU1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = L1 * cos(thetaU1);y0 = L1 * sin(thetaU1);r = L2;thetaStart = thetaU1 + thetaL2;thetaEnd = thetaU1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);x0 = L1 * cos(thetaL1);y0 = L1 * sin(thetaL1);r = L2;thetaStart = thetaL1 + thetaL2;thetaEnd = thetaL1;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);%-------------options.LineStyle = '--';options.Color='r';options.LineWidth = 0.5;x0 = 0;y0 = 0;r = sqrt(L1 * L1 + L2 * L2 + 2 * L1 * L2 * cos(thetaL2));thetaStart = 0;thetaEnd = 2 * pi;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)r = L1 + L2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options)x0 = L1 * cos(thetaU1);y0 = L1 * sin(thetaU1);r = L2;draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);x0 = L1 * cos(thetaL1);y0 = L1 * sin(thetaL1);draw_2d_arc(x0, y0, r, thetaStart, thetaEnd, options);xA1 = L1 * cos(thetaL1);yA1 = L1 * sin(thetaL1);xB1 = xA1 + L2 * cos(thetaL1 + thetaL2);yB1 = yA1 + L2 * sin(thetaL1 + thetaL2);xA2 = L1 * cos(thetaU1);yA2 = L1 * sin(thetaU1);xB2 = xA2 + L2 * cos(thetaU1 + thetaL2);yB2 = yA2 + L2 * sin(thetaU1 + thetaL2);xC1 = (L1 + L2) * cos(thetaL1);yC1 = (L1 + L2) * sin(thetaL1);xC2 = (L1 + L2) * cos(thetaU1);yC2 = (L1 + L2) * sin(thetaU1);plot([0, xA1, xB1], [0, yA1, yB1], 'lineStyle', '-', 'color', 'k', 'lineWidth', 3);plot([0, xA2, xB2], [0, yA2, yB2], 'lineStyle', ':', 'color', 'k', 'lineWidth', 3);fontsize = 15;delta = 25;text(0, 0, 'O', 'fontsize', fontsize);text(xA1, yA1 - delta, 'A_1', 'fontsize', fontsize);text(xB1, yB1 + delta, 'B_1', 'fontsize', fontsize);text(xA2, yA2 + delta, 'A_2', 'fontsize', fontsize);text(xB2, yB2 - delta, 'B_2', 'fontsize', fontsize);text(xC1, yC1, 'C_1', 'fontsize', fontsize);text(xC2, yC2, 'C_2', 'fontsize', fontsize);title('Workspace sketch in left handcoor', 'fontsize', 16); end set(gcf, 'color', 'w'); axis off; end clc; clear; close all; L1 = 200; L2 = 200; thetaLimit1 = [-135, 135] * pi / 180; thetaLimit2 = [-145, 145] * pi / 180;%% 畫工作空間 figure(1); handcoor = 0; draw_scara_workspace(L1, L2, thetaLimit1, thetaLimit2, handcoor) figure(2); handcoor = 1; draw_scara_workspace(L1, L2, thetaLimit1, thetaLimit2, handcoor)%% 畫工作空間草圖 figure(3); handcoor = 0; draw_scara_workspace_sketch(L1, L2, thetaLimit1, thetaLimit2, handcoor) figure(4); handcoor = 1; draw_scara_workspace_sketch(L1, L2, thetaLimit1, thetaLimit2, handcoor)

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