There　is　often　such　a　situation　that　the　condition　number　of　the　optimization　results　based　on　the　directional　manipulability　is　very　large,　which　is　not　conductive　for　control.　To　solve　this　problem,　a　new　dexterity　index-directional　manipulability　constrained　by　the　condition　number　is　proposed.　The　changing　pattern　of　the　partial　derivative　of　this　index　with　the　increase　of　the　condition　number　is　analyzed.　When　the　condition　number　is　small,　the　index　is　mainly　influenced　by　the　directional　manipulability.　While　the　condition　number　increases,　the　impact　of　the　directional　manipulability　on　the　index　declines,　and　the　impact　of　the　condition　number　increases.　Through　changing　the　partial　derivative,　the　condition　number　of　the　optimization　results　based　on　this　index　is　less　than　the　maximum　of　the　task　requirements.　Moreover,　the　directional　manipulability　can　be　retained　to　maximum.　The　simulation　researches　for　planar　3R　and　BUAA-RR　robot　and　the　experiment　research　for　planar　3R　manipulator　demonstrate　the　superiority　of　this　new　index.　Besides,　this　new　index　is　used　for　tolerant　performance　of　redundant　robots.　When　one　robot＇s　joint　fails　at　any　moment　during　the　task,　the　reduced　robot　based　on　this　new　index　has　a　good　manipulability　on　the　task　direction.　A　small　condition　number　of　the　reduced　robot　also　can　be　assured,　which　is　benefit　for　singularity　avoidance.　©　2010　Journal　of　Mechanical　Engineering.