m1=1; m2=1; l1=1;l2=1;g=1

eqs= {(m1+m2)*l1*a''[t]+m2*l2*Cos[a[t]-b[t]]*b''[t]+m2*l2*b'[t]*b'[t]*Sin[a[t]-b[t]]+(m1+m2)*g*Sin[a[t]]==0,m2*l2*b''[t]+m2*l1*a''[t]*Cos[a[t]-b[t]]-m2*l1*a'[t]*a'[t]*Sin[a[t]-b[t]]+m2*g*Sin[b[t]]==0}

condinit={a[0]==Pi/2.0,b[0]==Pi/2.0,a'[0]==0,b'[0]==0}

solution=NDSolve[{eqs,condinit},{a,b},{t,0,1000}]

Plot[Evaluate[{a[t],b[t]}/.solution],{t,0,100}]

c1[t_]=Evaluate[{Sin[a[t]],-1.0*Cos[a[t]]}/.solution]

c2[t_]=Evaluate[{Sin[a[t]]+Sin[b[t]],-1.0*Cos[a[t]]-1.0*Cos[b[t]]}/.solution]

Plot[{c1[t],c2[t]},{t,0,100}]

ParametricPlot[{c1[t],c2[t]},{t,0,100},PlotRange->{{-2,2},{-2.5,0}}]

Animate[Graphics[{{PointSize[.025],{Blue,Point[c1[t]]},{Orange,Point[c2[t]]},Line[{{0,0},Flatten[c1[t]],Flatten[c2[t]]}]},{Thickness[0.0002],{Orange,Line[Map[Function[c2[#]],Range[0,t,0.01]]]}}},PlotRange->{{-2,2},{-2.5,1}},Axes->True,Ticks->False,ImageSize->300],{t,0,100,0.01},AnimationRate->1.,SaveDefinitions->True]