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+import solids;
+import tube;
+import graph3;
+import palette;
+size(8cm);
+
+currentprojection=perspective(
+camera=(13.3596389245356,8.01038090435314,14.4864483364785),
+up=(-0.0207054323419367,-0.00472438375047319,0.0236460907598947),
+target=(-1.06042550499095,2.68154529985845,0.795007562120261));
+
+defaultpen(fontsize(6pt));
+
+// draw coordinates and frames
+// axis1 is defined by z axis of TBase
+// axis2 is defined by z axis of TEnd
+void DrawFrame(transform3 TBase, transform3 TEnd, string s)
+{
+ triple p1,v1,p2,v2;
+ p1=TBase*O;
+ v1=TBase*Z-p1;
+ p2=TEnd*O;
+ v2=TEnd*Z-p2;
+ triple n=cross(v1,v2);
+
+ real[][] A=
+ {
+ {v1.x,-v2.x,-n.x},
+ {v1.y,-v2.y,-n.y},
+ {v1.z,-v2.z,-n.z}
+ };
+
+ triple vb=p2-p1;
+
+ real[] b={vb.x,vb.y,vb.z};
+
+ // Get the extention along vector v1 and v2,
+ // so, we can get the common normal between two axis
+ real[] x=solve(A,b);
+
+ real s1=x[0];
+ real s2=x[1];
+
+ // get foot of a perpendicular on both axies
+ triple foot1=p1+s1*v1;
+ triple foot2=p2+s2*v2;
+
+ // draw two axis
+ triple axis_a,axis_b;
+ axis_a=p1+s1*v1*1.5;
+ axis_b=p1-s1*v1*1.5;
+ draw(axis_a--axis_b);
+
+ axis_a=p2+s2*v2*1.5;
+ axis_b=p2-s2*v2*1.5;
+ draw(axis_a--axis_b);
+
+ // draw "a"(common normal)
+ draw(Label("$a_{"+s+"}$"),foot1--foot2,linewidth(1pt));
+
+ // draw the coordinates frame
+ triple dx,dy,dz,org;
+ real length=0.8;
+
+ org=foot1;
+ dx =length*unit(foot2-foot1); // define the x axis of the frame on "a"
+ dz =length*unit(v1); // define the z axis which is along axis1
+ dy =length*unit(cross(dz,dx));
+
+ draw(Label("$X_{"+s+"}$",1,align=-dy-dz),org--(org+dx),red+linewidth(1.5pt),
+ Arrow3(8));
+ draw(Label("$Y_{"+s+"}$",1,align=2dy-dz-dx),org--(org+dy),
+ green+linewidth(1.5pt), Arrow3(8));
+ draw(Label("$Z_{"+s+"}$",1,align=-2dx-dy),org--(org+dz),
+ blue+linewidth(1.5pt), Arrow3(8));
+
+ dot(Label("$O_{"+s+"}$",align =-dx-dz,black),org,black); // origin
+
+}
+
+void DrawLink(transform3 TBase, transform3 TEnd, pen objStyle,string s)
+{
+ real h=1;
+ real r=0.5;
+ path3 generator=(0.5*r,0,h)--(r,0,h)--(r,0,0)--(0.5*r,0,0);
+ revolution vase=revolution(O,generator,0,360);
+ surface objSurface=surface(vase);
+
+ render render=render(merge=true);
+
+ // draw two cylinders
+ draw(TBase*objSurface,objStyle,render);
+ draw(TEnd*shift((0,0,-h))*objSurface,objStyle,render);
+
+ // draw the link between two cylinders
+ triple pStart=TBase*(0.5*h*Z);
+ triple pEnd =TEnd*(-0.5*h*Z);
+ triple pControl1=0.25*(pEnd-pStart)+TBase*(0,0,h);
+ triple pControl2=-0.25*(pEnd-pStart)+TEnd*(0,0,-h);
+ path3 p=pStart..controls pControl1 and pControl2..pEnd;
+ draw(tube(p,scale(0.2)*unitsquare),objStyle,render);
+}
+
+// t1 and t2 define the starting frame and ending frame of the first link(i-1)
+transform3 t1=shift((0,0,1));
+transform3 t2=shift((0,0,-1))*rotate(-20,Y)*shift((0,3,2));
+// as, the two links were connected, so t2 is also the starting frame of link(i)
+// t3 defines the ending frame of link(i)
+transform3 t3=t2*rotate(40,Z)*shift((0,3,1.5))*rotate(-15,Y)*shift(-1.5*Z);
+
+// draw link(i-1)
+DrawLink(t1,t2,palegreen,"i-1");
+DrawFrame(t1,t2,"i-1");
+// draw link(i)
+DrawLink(t2,t3,lightmagenta,"i");
+DrawFrame(t2,t3,"i");
+
+
+// draw angle alpha, which is the angle between axis(i-1) and axis(i)
+triple p0=(0,0,-1);
+triple p1=(0,0,2.3);
+triple p2=shift((0,0,-1))*rotate(-20,Y)*(0,0,4);
+draw(p0--p2,cyan);
+draw("$\alpha_{i-1}$",arc(p0,p1,p2,Y,CW),ArcArrow3(3));
+
+
+// draw angle theta, which is the angle between a_i and a_{i-1}
+transform3 tx=shift((0,0,-1))*rotate(-20,Y)*shift((0,3,0));
+p0=tx*O;
+p1=tx*(0,3,0);
+p2=tx*rotate(40,Z)*(0,3,0);
+draw(p0--p1,cyan);
+draw(p0--p2,cyan);
+
+triple p1a=tx*(0,1.5,0);
+draw("$\theta_{i}$",arc(p0,p1a,p2),ArcArrow3(3));
+
+// draw d_{i-1}
+triple org_i =t2*shift((0,0,1.5))*O;
+draw(Label("$d_{i}$",0.13),p0--org_i,linewidth(1pt));