Topological and Kinematic Singularities for  a Class of Parallel Mechanisms

<table>Constrained work space <svg height="16.025" id="M224" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 25 16.025" width="25" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.737)"><path d="M455 24l2 -2q66 26 128 97t139 204q126 219 183 291q58 73 113 73q22 0 36 -13t14 -34q0 -18 -12.5 -30.5t-30.5 -12.5q-43 0 -43 50h-1q-27 -7 -80 -88q-61 -92 -166 -273q-175 -301 -319 -301q-34 0 -46 4q85 79 141 233q44 124 99 214q56 93 138 179l-1 1
q-97 -16 -262 -312q-177 -319 -326 -319q-19 0 -47 6q30 24 59 72q26 42 44 85q21 49 44 111q56 151 192 287l-2 2q-17 -5 -36 -5q-48 0 -125 41q-76 42 -106 42q-47 0 -77.5 -29.5t-30.5 -71.5t24.5 -68t65.5 -26q37 0 65 20.5t56 65.5l16 -9q-54 -107 -139 -107
q-56 0 -92.5 35.5t-36.5 90.5q0 59 46 105t117 46q49 0 146 -49l36 -19q19 -12 40 -12q66 0 124 49l27 22l15 -16q-116 -115 -196 -321q-79 -205 -196 -310l2 -3q50 18 104 74q52 54 88 108q34 53 75 126q183 324 309 324q16 0 51 -11q-43 -31 -77 -96l-30 -57l-35 -79
l-31 -74q-92 -207 -195 -310z" id="x1D4B2"></path></g>
<g transform="matrix(.012,-0,0,-.012,18.388,15.825)"><path d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123
t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" id="x1D703"></path></g>
</svg> for <svg height="11.65" id="M225" style="vertical-align:-2.79825pt" version="1.1" viewbox="0 0 9.5749998 11.65" width="9.5749998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,8.1)"><path d="M20 461h192v-69q25 44 55.5 62.5t75.5 18.5q86 0 133.5 -58t47.5 -166q0 -119 -52 -190q-51 -72 -140 -72q-34 0 -58 12.5t-60 47.5v-126q0 -53 17 -76.5t60 -25.5v-24h-272v24q35 6 45.5 18t10.5 45v490q0 33 -11 46.5t-44 18.5v24zM214 346v-239q27 -65 77 -65
q85 0 85 192q0 182 -76 182q-37 0 -68 -37q-18 -21 -18 -33z" id="x1D429"></path></g>
</svg> with fixed <svg height="12.4375" id="M226" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 8.5874996 12.4375" width="8.5874996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M475 507q0 -83 -20 -172t-56 -167.5t-93.5 -129t-125.5 -50.5q-157 0 -157 227q0 78 21.5 164t59 161t96.5 123.5t126 48.5q79 0 114 -58t35 -147zM391 522q0 155 -81 155q-62 0 -111 -82.5t-73 -200.5h253q12 81 12 128zM373 346h-255q-12 -91 -12 -150q0 -72 20 -123
t63 -51q34 0 64 28.5t52.5 77t39 103t28.5 115.5z" id="x1D703"></path></g>
</svg>.</table>

Mathematical Problems in Engineering

fig6

Figure 6

Figure 6: Topological and Kinematic Singularities for  a Class of Parallel Mechanisms 