Advanced Methods for Robot-Environment Interaction towards an Industrial Robot Aware of Its Volume

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</svg> is the centre point, together with normalized and positive-oriented side vectors <svg height="14.0125" id="M24" style="vertical-align:-0.1911pt" version="1.1" viewbox="0 0 16.8125 14.0125" width="16.8125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,9.512,5.55)"><path d="M538 43v-23l-63 -7q-80 -9 -132 -26v65q-39 -37 -71.5 -51.5t-73.5 -14.5q-63 0 -98 38t-35 107v241q0 35 -9 47t-40 18v24h188v-334q0 -76 56 -76q38 0 82 47v276q0 34 -12.5 47t-48.5 16v24h200v-356q0 -34 12.5 -47.5t44.5 -14.5z" id="x1D42E"></path></g>
</svg><b> ,  </b><svg height="14.0125" id="M25" style="vertical-align:-0.1911pt" version="1.1" viewbox="0 0 16.15 14.0125" width="16.15" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,13.712)"><path d="M17 676h194v-259q53 56 122 56q82 0 135 -65t53 -169q0 -113 -63 -183t-163 -70q-71 0 -132 57l-79 -56h-12v607q0 30 -10 40.5t-45 17.5v24zM211 360v-263q0 -36 19 -57.5t50 -21.5q94 0 94 203q0 198 -93 198q-47 0 -70 -59z" id="x1D41B"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.512,5.55)"><path d="M334 461h151v-24q-24 -2 -34 -11.5t-24 -43.5l-160 -396h-26l-120 296q-47 116 -66 141q-7 9 -34 14v24h250v-24l-18 -2q-35 -5 -35 -24q0 -9 9 -31l79 -203l72 182q11 27 11 45t-11 24.5t-44 8.5v24z" id="x1D42F"></path></g>
</svg><b> , </b>and <svg height="14.0125" id="M26" style="vertical-align:-0.1911pt" version="1.1" viewbox="0 0 24.450001 14.0125" width="24.450001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,13.712)"><path d="M17 676h194v-259q53 56 122 56q82 0 135 -65t53 -169q0 -113 -63 -183t-163 -70q-71 0 -132 57l-79 -56h-12v607q0 30 -10 40.5t-45 17.5v24zM211 360v-263q0 -36 19 -57.5t50 -21.5q94 0 94 203q0 198 -93 198q-47 0 -70 -59z" id="x1D41B"></path></g>
<g transform="matrix(.012,-0,0,-.012,9.512,5.55)"><path d="M572 461h135v-24q-19 -4 -29.5 -14.5t-20.5 -35.5l-155 -401h-23l-102 310l-125 -310h-24l-148 374q-21 49 -29 60q-8 12 -28 17v24h222v-24q-38 -5 -38 -28q0 -11 39 -115l45 -118l68 171q-5 13 -6 19q-13 47 -22 58q-7 10 -37 13v24h234v-24q-28 -2 -38 -7t-10 -17
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</svg>. The vectors <svg height="16.450001" id="M27" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 15.925 16.450001" width="15.925" 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="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
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<g transform="matrix(.012,-0,0,-.012,8.837,16.25)"><path d="M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
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</svg><b> ,  </b><svg height="16.450001" id="M28" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 15.225 16.450001" width="15.225" 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="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
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<g transform="matrix(.012,-0,0,-.012,8.837,16.25)"><path d="M457 332q0 -81 -41 -161.5t-105.5 -131.5t-129.5 -51q-46 0 -79 30.5t-33 86.5q0 18 7 51q18 95 46 187q6 19 6 33q0 7 -7 7q-24 0 -78 -64l-20 23q32 48 73 77t78 29q33 0 33 -46q0 -38 -10 -70q-28 -92 -39 -152q-7 -38 -7 -57q0 -78 69 -78q68 0 118 75.5t50 194.5
q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" id="x1D463"></path></g>
</svg><b> , </b>and <svg height="16.450001" id="M29" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 17.987499 16.450001" width="17.987499" 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="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" id="x210E"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.837,16.25)"><path d="M689 332q0 -81 -40 -161.5t-103.5 -131.5t-127.5 -51q-32 0 -58.5 16t-40.5 46q-70 -62 -139 -62q-45 0 -77.5 30t-32.5 86q0 8 7 52q11 77 45 187q6 19 6 34q0 6 -7 6q-24 0 -78 -64l-20 23q32 48 73 77t78 29q32 0 32 -43q0 -30 -12 -73q-27 -95 -38 -152
q-8 -44 -8 -58q0 -77 68 -77q31 0 60 33.5t39 77.5l62 260l75 16l5 -6l-63 -246q-8 -29 -8 -58q0 -77 68 -77q66 0 116 75.5t50 194.5q0 43 -12 57q-11 12 -11 24q0 19 15 35.5t34 16.5q18 0 30.5 -34.5t12.5 -81.5z" id="x1D464"></path></g>
</svg> are the distances from the centre of the box to the centre of its faces.</table>

Journal of Robotics

fig5

Figure 5

Figure 5: Advanced Methods for Robot-Environment Interaction towards an Industrial Robot Aware of Its Volume 