This paper presents a pipe cutting Robot system with two di erent cutting methods: the method with the end-e ector moves on cutting path and direction while the stationary pipe and the method with the end-e ector moves on a straight line while the rotating pipe to create the desired cutting path and direction.
❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r ❈♦♥tr♦❧ ♦❢ P✐♣❡ ❈✉tt✐♥❣ ❘♦❜♦t✿ ❆ ▼♦r❡ ❊❢❢❡❝t✐✈❡ ▼❡t❤♦❞ ◗✉♦❝ ❇❛♦ ❉■❊P ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❱❙❇✕❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❖str❛✈❛✱ ✶✼✳ ❧✐st♦♣❛❞✉ ✶✺✱ ✼✵✽ ✸✸ ❖str❛✈❛✱ ❈③❡❝❤ ❘❡♣✉❜❧✐❝ ❞✐❡♣q✉♦❝❜❛♦❅❣♠❛✐❧✳❝♦♠ ✭❘❡❝❡✐✈❡❞✿ ✵✹✲❆✉❣✉st✲✷✵✶✼❀ ❛❝❝❡♣t❡❞✿ ✵✹✲❖❝t♦❜❡r✲✷✵✶✼❀ ♣✉❜❧✐s❤❡❞✿ ✸✵✲◆♦✈❡♠❜❡r✲✷✵✶✼✮ ❚❤✐s ♣❛♣❡r ♣r❡s❡♥ts ❛ ♣✐♣❡ ❝✉tt✐♥❣ ❘♦❜♦t s②st❡♠ ✇✐t❤ t✇♦ ❞✐✛❡r❡♥t ❝✉tt✐♥❣ ♠❡t❤♦❞s✿ t❤❡ ♠❡t❤♦❞ ✇✐t❤ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ♦♥ ❝✉t✲ t✐♥❣ ♣❛t❤ ❛♥❞ ❞✐r❡❝t✐♦♥ ✇❤✐❧❡ t❤❡ st❛t✐♦♥❛r② ♣✐♣❡ ❛♥❞ t❤❡ ♠❡t❤♦❞ ✇✐t❤ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ✇❤✐❧❡ t❤❡ r♦t❛t✐♥❣ ♣✐♣❡ t♦ ❝r❡❛t❡ t❤❡ ❞❡s✐r❡❞ ❝✉tt✐♥❣ ♣❛t❤ ❛♥❞ ❞✐r❡❝t✐♦♥✳ ❚❤❡ ❝✉tt✐♥❣ tr❛❥❡❝t♦r② ❛r❡ ❞❡s❝r✐❜❡❞✱ t❤❡ ❘♦❜♦t ♠♦❞❡❧ ✐s ❝♦♥✲ str✉❝t❡❞✱ s♦❧✈✐♥❣ t❤❡ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s✱ ♣❧❛♥✲ ♥✐♥❣ t❤❡ tr❛❥❡❝t♦r② ♦❢ ♠♦t✐♦♥✱ s✐♠✉❧❛t✐♥❣ ❛♥❞ ❝♦♥tr♦❧❧✐♥❣ ❘♦❜♦t ✐♥ ▼❛t❧❛❜✱ ❛♥❞ ❞❡s✐❣♥✐♥❣ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❘♦❜♦t t♦ ✈❡r✐❢②✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤❡ t✇♦ ♠❡t❤♦❞s ❛r❡ ❝♦♠♣❛r❡❞ t♦ ♣♦✐♥t ♦✉t ❛ ❜❡tt❡r ♦♥❡✳ ❚❤✐s r❡s❡❛r❝❤ ❜✉✐❧❞s ✉♣ ❛♥ ✐♠♣♦rt❛♥t ❢♦✉♥✲ ❞❛t✐♦♥ ❢♦r ❝❤♦♦s✐♥❣ ❛♥ ❡✛❡❝t✐✈❡ ♠❡t❤♦❞ ❢♦r ♣✐♣❡ ❝✉tt✐♥❣ ❘♦❜♦t ✐♥ ✐♥❞✉str②✳ ❆❜str❛❝t✳ ❑❡②✇♦r❞s ❈✉tt✐♥❣ r♦❜♦t✱ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s✱ ❝✉tt✐♥❣✱ ❝♦♥tr♦❧✱ r♦❜♦t tr❛❥❡❝t♦r② ♥✐♥❣✳ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ♣✐♣❡ ♣❧❛♥✲ t❤❡ tr❛❥❡❝t♦r② ❛♥❞ ❞✐r❡❝t✐♦♥ ❝❤❛♥❣❡ ❝♦♥t✐♥✉✲ ♦✉s❧②✳ ❲✐t❤ t❤❛t r❡q✉✐r❡♠❡♥t✱ ❝♦♥✈❡♥t✐♦♥❛❧ t♦♦❧s ❝❛♥♥♦t ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ❛♥❞ t❤❡ ❛♣♣❧✐❝❛✲ t✐♦♥ ♦❢ r♦❜♦ts ✐s ♥❡❝❡ss❛r②✳ ❚❤❡ ✉s❡ ♦❢ r♦❜♦t ❢♦r ❝✉tt✐♥❣ ♣✐♣❡ ❤❛s ❜❡❝♦♠❡ ✈❡r② ♣♦♣✉❧❛r ✐♥ t❤❡ ✇♦r❧❞✳ ■♥ ❱✐❡t♥❛♠✱ t❤✐s t❡❝❤✲ ♥✐q✉❡ ❤❛s ♥♦t ②❡t ❜❡❡♥ ✇✐❞❡❧② ❛♣♣❧✐❡❞✱ ❛♥❞ t❤❡r❡ ❛r❡ ❢❡✇ s❝✐❡♥t✐✜❝ ♣✉❜❧✐❝❛t✐♦♥s ✐♥ t❤✐s ✜❡❧❞✳ ■♥ ❬✶❪✱ t❤❡ ❛✉t❤♦rs ♠❡♥t✐♦♥❡❞ ❛ ❉❡❧t❛ ❘♦❜♦t ❢♦r ❝✉tt✐♥❣ ❤✐❣❤✲s♣❡❡❞ ❧❛s❡r ✇✐t❤ t❤❡ ♥✉♠❡r✲ ♦✉s ❛❞✈❛♥t❛❣❡s ♦❢ r♦❜♦t✿ ❤✐❣❤❡r st✐✛♥❡ss✱ ❢❡✇❡r ❥♦✐♥ts✱ t❤❡ ❛❜✐❧✐t② ♦❢ tr❛♥s♣♦rt✐♥❣ ❤❡❛✈✐❡r ❧♦❛❞s✱ ❛♥❞ ❤✐❣❤❡r ❛❝❝✉r❛❝②✳ ❚❤❡ ♠❛✐♥ ❞r❛✇❜❛❝❦ ✐s t❤❡ s♠❛❧❧ ✇♦r❦s♣❛❝❡✱ ❛♥❞ t❤✐s ♣❛♣❡r ❛❧s♦ ❞♦❡s ♥♦t ♠❡♥t✐♦♥ ♠✉❝❤ ❛❜♦✉t t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ❉❡❧t❛ ❘♦❜♦t t♦ ❝✉t st❡❡❧ ♣✐♣❡s✳ ■♥ ❬✷❪✱ t❤❡ ❛✉t❤♦rs ♣r❡s❡♥t❡❞ ❛ ♣✐♣❡ ❝✉tt✐♥❣ t❡❝❤♥✐q✉❡ t❤❛t ✐♥❝❧✉❞❡❞ ❛ ♣✐♣❡ ❝✉tt✐♥❣ ❘♦❜♦t t❤❛t t❤❡ r♦❜♦t ❛r♠ ♠♦✈❡s ❛♥❞ t❤❡ ♣✐♣❡ ✐s st❛✲ t✐♦♥❛r② ❞✉r✐♥❣ ❝✉tt✐♥❣✳ ❚❤❡ ❛✉t❤♦rs s✉❝❝❡ss❢✉❧❧② ❜✉✐❧❞s ✸❉ s✐♠✉❧❛t✐♦♥ ❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ♠♦❞❡❧✱ s♦❧✈✐♥❣ t❤❡ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s✱ ♣❧❛♥♥✐♥❣ t❤❡ tr❛✲ ❥❡❝t♦r② ❛s ✇❡❧❧ ❛s ❞❡s✐❣♥s ❘♦❜♦t ❝♦♥tr♦❧❧❡r✳ ❚❤❡ r❡s✉❧ts ♦❢ s✐♠✉❧❛t❡❞ ❛♥❞ ❡①♣❡r✐♠❡♥t❛❧ ❡rr♦rs ❛r❡ ♣r♦✈✐❞❡❞✳ ❍♦✇❡✈❡r✱ t❤❡ ❛✉t❤♦rs ♦♥❧② st♦♣♣❡❞ ❛t t❤❡ ♠❡t❤♦❞ ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ✇❤✐❧❡ t❤❡ st❛t✐♦♥❛r② ♣✐♣❡ ✇✐t❤♦✉t ♠❡♥t✐♦♥✐♥❣ t❤❡✐r ❝♦♦r✲ ❞✐♥❛t❡❞ ♠♦t✐♦♥✳ ■♥ ✐♥❞✉str②✱ ❣❛s❡s ❛♥❞ ❧✐q✉✐❞s ❛r❡ tr❛♥s♣♦rt❡❞ ❞❛✐❧② ❜② ♣✐♣❡❧✐♥❡s ❛♥❞ t❤❡s❡ ♣✐♣❡❧✐♥❡ s②st❡♠s ♣❛✐r❡❞ t♦❣❡t❤❡r ✐♥ ❛ ❝♦♠♣❧❡① ✇❛②✳ ❚♦ ❝r❡❛t❡ ■♥ t❤✐s ♣❛♣❡r✱ t❤❡ ❛✉t❤♦r ♣r❡s❡♥ts ❛♥♦t❤❡r t❤❡♠✱ t❤❡ st❡❡❧ ♣✐♣❡s ❛r❡ ❝✉t ❛♥❞ ✇❡❧❞❡❞ t♦✲ ♠❡t❤♦❞ ♦❢ ❝✉tt✐♥❣ ♣✐♣❡ ♠♦r❡ ❡✛❡❝t✐✈❡❧② ✇✐t❤ ❛ ✻ ❣❡t❤❡r✳ ❍♦✇❡✈❡r✱ t❤✐s ✐s ♥♦t s✐♠♣❧❡ ❜❡❝❛✉s❡ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ♣✐♣❡ ❝✉tt✐♥❣ ❘♦❜♦t✱ ❝♦♥s✐st✲ t❤❡r❡ ❛r❡ ❝♦♠♣❧❡① ❥♦✐♥ts t❤❛t r❡q✉✐r❡ ❝✉tt✐♥❣ ✐♥❣ ♦❢ ✺ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ r♦❜♦t ❛r♠ ❛♥❞ t❤❡ ❛♥❞ ✇❡❧❞✐♥❣ ♣❛t❤s t♦ ❜❡ ❝♦♠♣❧✐❝❛t❡❞ ✐♥ ✇❤✐❝❤ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✾✺ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ❝r❡❛t❡❞ ❜② t❤❡ r♦t❛t✐♥❣ ♠♦t✐♦♥ ❲❤❡♥ t✉r♥✐♥❣ ♣✐♣❡ R2 ❛♥ ❛♥❣❧❡ ar◦ ❛r♦✉♥❞ ♦❢ ❛ ♣✐♣❡✳ ❲✐t❤ t❤✐s ♠❡t❤♦❞✱ t❤❡ ❡♥❞✲❡✛❡❝t♦r X−1 ❛①✐s ✭❢♦r ❡①❛♠♣❧❡ 90◦ ✱ s❡❡ ❋✐❣✳ ✶✮✱ ✇❡ ❤❛✈❡ ✇✐❧❧ ♦♥❧② ♠♦✈❡ ♦♥ ❛ str❛✐❣❤t ❧✐♥❡✱ ❛♥❞ t❤❡ ♣✐♣❡ ❊q✳ ✭✷✮✿ ✇✐❧❧ r♦t❛t❡ ✐♥ ❝♦♥❥✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ♠♦✈❡♠❡♥t ♦❢ 0 x2 x2 t❤❡ ❡♥❞✲❡✛❡❝t♦r t♦ ❝r❡❛t❡ ❝✉tt✐♥❣ ♣❛t❤ ✐♥ r❡❛❧✐t②✳ y2 = 0 car −sar y2 , ✭✷✮ ❚❤✐s r❡s✉❧t ✐s ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ r❡s✉❧ts ✐♥ ❬✷❪ z2 sar car z2 t♦ ♣♦✐♥t ♦✉t t❤❛t t❤✐s ❝✉tt✐♥❣ ♠❡t❤♦❞ ✐s ❜❡tt❡r t❤❛♥ t❤❡ ❝✉tt✐♥❣ ♠❡t❤♦❞ ✐♥ ❬✷❪✳ ✇❤❡r❡ ✷✳ • ar✿ ❛♥❣❧❡ ❜❡t✇❡❡♥ R1 ❛♥❞ R2 ❛♥❞ P✐♣❡ ❈✉tt✐♥❣ Pr♦❜❧❡♠ ❚❤❡ ❝✉tt✐♥❣ ♣❛t❤s ❛♥❞ ❝✉tt✐♥❣ ❞✐r❡❝t✐♦♥s ❝❛♥ ❤❛♣♣❡♥ ♠❛♥② ❝❛s❡s✱ ❞❡♣❡♥❞✐♥❣ ♦♥ ♣✐♣❡❧✐♥❡ ❛s✲ s❡♠❜❧② ♣♦s✐t✐♦♥ ❛♥❞ ✇❡❧❞✐♥❣ ❝♦♥❞✐t✐♦♥s✳ ❚❤✐s ♣❛♣❡r ✇✐❧❧ ❢♦❝✉s ♦♥ ❍②♣❡r❜♦❧✐❝ P❛r❛❜♦❧♦✐❞ Pr✐♥❣❧❡s❀ ❛ ❝♦♠♠♦♥ ♣❛t❤ ✐s ❝r❡❛t❡❞ ❜② t✇♦ ✐♥✲ t❡rs❡❝t✐♥❣ ♣✐♣❡s ❛s s❤♦✇♥ ✐♥ ❋✐❣✳ ✶✳ ✷✳✶✳ ❈✉tt✐♥❣ P❛t❤ ■♥ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {−1}✱ ♣❧❛❝❡ ♣✐♣❡s R1 ❛♥❞ R2 ✇❤✐❝❤ ❤❛✈❡ ❝❡♥t❡r ❧✐♥❡s ❝♦✐♥❝✐❞❡♥t ✇✐t❤ Z−1 ❛①✐s✱ ❡❛❝❤ ♣✐♣❡ ❡q✉❛t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❊q✳ ✭✶✮✿ x2n + yn2 = Rn2 , − 12 Ln ≤ zn ≤ 12 Ln , ✭✶✮ • sar ✿ sin ar✱ car ✿ cos ar✳ ❇❛s❡❞ ♦♥ ❊q✳ ✭✶✮ ❛♥❞ ❊q✳ ✭✷✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♣✐♣❡ R2 ❛❢t❡r t✉r♥✐♥❣ ar◦ ❛s ❊q✳ ✭✸✮✿ x22 + (y2 car − z2 sar )2 = R22 , − 21 L2 ≤ y2 sar + z2 car ≤ 12 L2 ❋r♦♠ ❊q✳ ✭✶✮✱ ❊q✳ ✭✷✮ ❛♥❞ ❊q✳ ✭✸✮✱ ✇❡ ❤❛✈❡ t❤❡ ❧♦❝✉s ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ♣✐♣❡s ❛s ❊q✳ ✭✹✮✿ x2 + y = R12 , x2 + (ycar − zsar )2 = R22 R2 sϕ = x, R2 cϕ = ycar − zsar , • L✱ R✿ t❤❡ ❧❡♥❣t❤ ❛♥❞ r❛❞✐✉s ♦❢ t✇♦ ♣✐♣❡s✱ • x✱ y ✱ z ✿ ❝♦♦r❞✐♥❛t❡s ♦❢ t✇♦ ♣✐♣❡s ❛♥❞ ac R2 ar Z Y ✇❤❡r❡ ≤ ϕ ≤ 2π ✱ sϕ ✿ sin ϕ✱ cϕ ✿ cos ϕ✳ x = R2 sϕ , ✭✻✮ y = ± R12 − (R2 sϕ )2 , ✭✼✮ DETAIL B {0} X B Y Z X z= {-1} ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ♣✐♣❡s✳ ✭✺✮ ❇❛s❡❞ ♦♥ ❊q✳ ✭✹✮ ❛♥❞ ❊q✳ ✭✺✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❧♦✲ ❝✉s ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ♣✐♣❡s ✐♥ t❤❡ ❝♦♦r❞✐✲ ♥❛t❡ ❢r❛♠❡ {−1} ❛s ❊q✳ ✭✻✮✱ ❊q✳ ✭✼✮ ❛♥❞ ❊q✳ ✭✽✮✿ • n = 1✿ ♣✐♣❡ R1 ✱ n = 2✿ ♣✐♣❡ R2 ✳ ✾✻ ✭✹✮ ❆ss✉♠✐♥❣ t❤❛t R1 ≥ R2 ✱ s❡t✿ ✇❤❡r❡ ❋✐❣✳ ✶✿ ✭✸✮ R1 −R2 cϕ ± car R12 − (R2 sϕ )2 sar ✭✽✮ ❙❡t ✉♣ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} s♦ t❤❛t t❤❡ X0 ❛①✐s ✐s ❝♦✐♥❝✐❞❡❞ ❛♥❞ r❡✈❡rs❡❞ ✇✐t❤ t❤❡ Z−1 ❛①✐s✳ ❚❤❡ ❞✐st❛♥❝❡ ❢r♦♠ {−1} t♦ {0} ✐s d0 ✭s❡❡ ❋✐❣✳ ✶✮✳ ❚❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} ✐s ✜①❡❞ ❛♥❞ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {−1} ❝❛♥ r♦t❛t❡ ❛r♦✉♥❞ Z0 ❛♥ ❛♥❣❧❡ θ0 ✳ ❚❤❡ ❧♦❝✉s ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t✇♦ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ♣✐♣❡s R1 ❛♥❞ R2 ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} ✐s ❣✐✈❡♥ ❜② ❊q✳ ✭✾✮✱ ❊q✳ ✭✶✵✮✱ ❊q✳ ✭✶✶✮ ❛♥❞ ✭✶✷✮✿ x Q1x 0 −1 d0 −c0 s0 0 y = Q1y , Q1 = s0 c0 z Q1z 1 0 ✭✾✮ ✇❤✐❝❤ ♠❛♥② ❞✐r❡❝t✐♦♥s t♦ ❣♦ t❤r♦✉❣❤✱ ❜✉t ♦♥❧② ♦♥❡ ❞✐r❡❝t✐♦♥ ✐s r❡❛s♦♥❛❜❧❡ ✇✐t❤ t❤❡ r❡q✉✐r❡✲ ♠❡♥ts ❛❜♦✉t ♣✐♣❡ ✇❡❧❞✐♥❣ ❝♦♥❞✐t✐♦♥s ❬✸❪✱ ❞❡✲ ♣❡♥❞✐♥❣ ♦♥ t❤❡ ❝✉tt✐♥❣ ❛♥❣❧❡ ❬✹❪ ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣♦✐♥t✱ s❤♦✇♥ ✐♥ ❋✐❣✳ ✷✳ ❈✉tt✐♥❣ ❞✐r❡❝t✐♦♥ ❝❤❛♥❣❡s ❝♦♥t✐♥✉♦✉s❧② ❞✉r✐♥❣ ❝✉tt✐♥❣ ♣r♦❝❡ss✳ ✇❤❡r❡ M-CY-Z −R2 cϕ ± car sar α , e ✭✶✵✮ ❲❤❡♥ ❝✉tt✐♥❣✱ ♣✐♣❡ R2 ♠✉st ❜❡ r❡✲❛♣♣❧✐❡❞ ❛♥ ❛♥❣❧❡ −ar◦ s♦ t❤❛t t❤❡ ❝❡♥t❡r ❧✐♥❡ ♦❢ R2 ❝♦✐♥✲ ❝✐❞❡s ✇✐t❤ Z−1 ❛①✐s✳ ❚❤❡ ❝✉tt✐♥❣ ♣❛t❤ ♦❢ ♣✐♣❡ R2 ✐s ❣✐✈❡♥ ❜② ❊q✳ ✭✶✸✮✱ ❊q✳ ✭✶✹✮✱ ❊q✳ ✭✶✺✮ ❛♥❞ ❊q✳ ✭✶✻✮✳ −1 Q1x Q2x Q2 = 0 c−ar −s−ar Q1y = Q2y , s−ar c−ar Q1z Q2z ✭✶✸✮ sar R1 ❈✉tt✐♥❣ ❞✐r❡❝t✐♦♥ ♦❢ ♣✐♣❡ R1 ✳ α = Y−1 O−1 Q1 + ac, , ✭✶✹✮ ✭✶✼✮ Y−1 O−1 Q1 = arctan2( Q21x + Q21z , Q1y ) ✭✶✽✮ R12 − (R2 sϕ )2 ) Q2y = car (−c0 R2 sϕ ± s0 + sar (s0 R2 sϕ ± c0 {-1} ■♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {−1}✱ ♣❧❛♥❡ (M, C) ❝♦♥t❛✐♥s Y−1 ❛①✐s ❛♥❞ ♣❛ss❡s ❝✉tt✐♥❣ ♣♦✐♥t Q1 ✳ ■♥ ♣❧❛♥❡ (M, C)✱ e ✐s t❤❡ ❧✐♥❡ t❤❛t ❝♦♥t❛✐♥s ❝✉t✲ t✐♥❣ ❞✐r❡❝t✐♦♥ ❛♥❞ ♣❛ss❡s ❝✉tt✐♥❣ ♣♦✐♥t Q1 ✱ α ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ e ❛♥❞ Y−1 ❛①✐s ✭❣✐✈❡♥ ❜② ❊q✳ ✭✶✼✮ ❛♥❞ ❊q✳ ✭✶✽✮✮✳ β ✐s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ♣❧❛♥❡ (M, C) ❛♥❞ (Y, Z)✳ ✇❤❡r❡ R12 − (R2 sϕ )2 Y X ❋✐❣✳ ✷✿ −R2 cϕ ± car Q1 ✭✶✷✮ R12 − (R2 sϕ )2 , s0 ✿ sin θ0 ✱ c0 ✿ cos θ0 ✳ Q2x = d0 − X-Y Z ✭✶✶✮ R12 − (R2 sϕ )2 , Q1y = −c0 R2 sϕ ± s0 Q1z = s0 R2 sϕ ± c0 − (R2 sϕ )2 ac Q1x = d0 − R12 | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r R12 − (R2 sϕ )2 ), ✭✶✺✮ Q2z = −sar (−c0 R2 sϕ ± s0 R12 − (R2 sϕ )2 ) ac✿ t❤❡ st❛♥❞❛r❞ ❝✉tt✐♥❣ ❛♥❣❧❡ ✐s ❣✐✈❡♥ ❜❡✲ ❢♦r❡ ❬✹❪ ✭s❡❡ ❋✐❣✳ ✶ ❛♥❞ ❋✐❣✳ ✷✮✳ e ✐s ❢♦✉♥❞ ❜② r♦t❛t✐♥❣ ❛♥ ✐♠❛❣✐♥❛r② ❧✐♥❡ t❤r♦✉❣❤ Q1 ❛♥❞ ♣❛r❛❧❧❡❧✐♥❣ Y−1 ❛①✐s ✇✐t❤ ❛♥ ❛♥✲ ❣❧❡ β ❛r♦✉♥❞ Y−1 ❛①✐s✱ ❛♥❞ t❤❡♥ r♦t❛t✐♥❣ t❤✐s ❧✐♥❡ ✇✐t❤ ❛♥ ❛♥❣❧❡ α ❛r♦✉♥❞ Q1 ✳ ❡ ✐s t❤❡ ❧✐♥❡ ❊q✉❛t✐♦♥ ✭✾✮✱ ❊q✳ ✭✶✵✮✱ ❊q✳ ✭✶✶✮✱ ❊q✳ ✭✶✷✮✱ ❊q✳ ✭✶✸✮✱ ❊q✳ ✭✶✹✮✱ ❊q✳ ✭✶✺✮ ❛♥❞ ❊q✳ ✭✶✻✮ ❞❡✲ ❝♦♥t❛✐♥✐♥❣ t❤❡ ❞✐r❡❝t✐♦♥ ✇❡ ♥❡❡❞✳ s❝r✐❜❡ t❤❡ ♣❛t❤s ✐♥ ✇❤✐❝❤ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ❚❤❡ ❝✉tt✐♥❣ ❞✐r❡❝t✐♦♥ ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ ♦♥ t❤❡♠ ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} ✇❤❡♥ ❝✉t✲ {−1} ✐s ❛ ✸①✸ ♠❛tr✐① ❛♥❞ ❣✐✈❡♥ ❜② ❊q✳ ✭✶✾✮✿ t✐♥❣✳ + car (s0 R2 sϕ ± c0 ✷✳✷✳ R12 − (R2 sϕ )2 ) ✭✶✻✮ ❈✉tt✐♥❣ ❉✐r❡❝t✐♦♥ H−1 ❇❡s✐❞❡ ❝✉tt✐♥❣ ♣❛t❤✱ ✇❡ ♠✉st ♣❛② ❛tt❡♥t✐♦♥ t♦ t❤❡ ❝✉tt✐♥❣ ❞✐r❡❝t✐♦♥✳ ❲✐t❤ ❡❛❝❤ ❝✉tt✐♥❣ ♣♦✐♥t ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ cβ = RY (α) RX (β) = −sβ sα sβ cα sα cβ cα sβ −sα cα cβ ✭✶✾✮ ✾✼ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r ❋r♦♠ ❊q✳ ✭✶✾✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❝✉tt✐♥❣ ❞✐r❡❝t✐♦♥ ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} ❛s ❊q✳ ✭✷✵✮✿ ❋r♦♠ ❊q✳ ✭✷✶✮✱ ✇❡ ♦❜t❛✐♥✿ H0 =−1 R H−1 sβ −sα cβ −c c −c s = β α sβ + s0 cα s0 cβ s0 sα sβ + c0 cα θ0 = arctan2(∓R2 cϕ , ✷✳✸✳ R12 − (R2 sϕ )2 ) ✭✷✷✮ −cα cβ −c0 cα sβ − s0 sα ❲❤✐❧❡ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ✐♥ ❊q✳ ✭✷✶✮ ✇✐t❤ ❞✐r❡❝t✐♦♥ ✐♥ ❊q✳ ✭✷✵✮✱ t❤❡ ♣✐♣❡ s0 cα sβ − c0 sα ✭✷✵✮ r♦t❛t❡ ❛♥ ❛♥❣❧❡ θ0 ✐♥ ❊q✳ ✭✷✷✮✳ ❚❤✐s ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ✇✐❧❧ ❝r❡❛t❡ t❤❡ ❍②♣❡r❜♦❧✐❝ P❛r❛❜♦❧♦✐❞ Pr✐♥❣❧❡s ✐♥ r❡❛❧✳ ❈✉tt✐♥❣ ▼❡t❤♦❞ ❚❤❡r❡ ❛r❡ t✇♦ ❣❡♥❡r❛❧ ❝❛s❡s ♦❢ ❝✉tt✐♥❣✿ st❛t✐❝ ♣✐♣❡ ✇❤✐❧❡ ♠♦✈✐♥❣ ❡♥❞✲❡✛❡❝t♦r ❬✷❪ ❛♥❞ r♦t❛t✲ ✐♥❣ ♣✐♣❡ ✇❤✐❧❡ ♠♦✈✐♥❣ ❡♥❞✲❡✛❡❝t♦r✳ ❊s♣❡❝✐❛❧❧②✱ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ♦♥ ❛ str❛✐❣❤t s❡❝t✐♦♥ ✇✐t❤ ❝✉tt✐♥❣ ❞✐r❡❝t✐♦♥ ❢♦❧❧♦✇✐♥❣ t♦ ❊q✳ ✭✷✵✮✳ ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r♦t❛t✐♥❣ ♣✐♣❡ ✇❤✐❧❡ ♠♦✈✲ ✐♥❣ ❡♥❞✲❡✛❡❝t♦r ♦♥ ❛ str❛✐❣❤t s❡❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❊q✳ ✭✷✶✮ ❛♥❞ ❊q✳ ✭✷✷✮✳ ■t ✇✐❧❧ ❝r❡❛t❡ ❍②♣❡r❜♦❧✐❝ P❛r❛❜♦❧♦✐❞ Pr✐♥❣❧❡s ❧✐❦❡ t❤❡ r❡❛❧✐t②✳ ❚❤✐s ❝❛s❡ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❋✐❣✳ ✸✳ ✸✳ ❘♦❜♦t ▼♦❞❡❧ ❚❤❡ ❘♦❜♦t ♠♦❞❡❧ ✉s❡❞ ✐♥ t❤✐s ♣❛♣❡r ✐♥❝❧✉❞❡s ✺ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ r♦❜♦t ❛r♠ ❛♥❞ ❛ ❞❡❣r❡❡ ♦❢ ❢r❡❡❞♦♠ ❝r❡❛t❡❞ ❜② t❤❡ r♦t❛t✐♥❣ ♠♦t✐♦♥ ♦❢ t❤❡ ♣✐♣❡✳ ❚❤✐s ♠♦❞❡❧ ✐s s❤♦✇♥ ✐♥ ❋✐❣✳ ✹✳ a3 a2 Y1 a4 Y2 X1 Y3 X3 X2 X4 d1 d5 Z4 X5 Z5 Z0 Y-1 X0 ❋✐❣✳ ✸✿ ❚r❛❥❡❝t♦r② ♦❢ ❡♥❞✲❡✛❡❝t♦r ✐s ❛ ♣❛r❛❧❧❡❧ str❛✐❣❤t s❡❝t✐♦♥ ✇✐t❤ X0 ❛①✐s✱ ❛♥❞ ✐s ❧♦❝❛t❡❞ r✐❣❤t ❛t t❤❡ t♦♣ ♦❢ t❤❡ ♣✐♣❡✳ ❚❤❡ str❛✐❣❤t s❡❝t✐♦♥ ❜❡❧♦♥❣s t♦ ♣❧❛♥❡ (X0 , Z0 )✱ ✐♥ ✇❤✐❝❤ Y0 = ❛♥❞ Z0 = R1 ✳ ❋r♦♠ ❊q✳ ✭✾✮✱ ❊q✳ ✭✶✵✮✱ ❊q✳ ✭✶✶✮ ❛♥❞ ❊q✳ ✭✶✷✮✱ ✇❡ ✐♥❢❡r✿ −R2 cϕ ± car R12 − (R2 sϕ )2 d0 − sar −1 Q1 = 2 −c0 R2 sϕ ± s0 R1 − (R2 sϕ ) s0 R2 sϕ ± c0 R12 − (R2 sϕ )2 −R2 cϕ ± car R12 − (R2 sϕ )2 d0 − sar = R1 ✭✷✶✮ ✾✽ Z -1 ❚❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ r♦t❛t✐♥❣ ♣✐♣❡ ✇❤✐❧❡ ♠♦✈✐♥❣ ❡♥❞✲❡✛❡❝t♦r ♦♥ ❛ str❛✐❣❤t s❡❝t✐♦♥✳ d0 ❋✐❣✳ ✹✿ ✸✳✶✳ ❚❤❡ ♠♦❞❡❧ ♦❢ ❘♦❜♦t✳ ❋♦r✇❛r❞ ❑✐♥❡♠❛t✐❝s ❚❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} ✐s t❤❡ ❣❧♦❜❛❧ ❝♦♦r❞✐✲ ♥❛t❡ ❢r❛♠❡ ♦❢ ❘♦❜♦t s②st❡♠✳ ❘♦❜♦t s②st❡♠ ✐s ❞✐✈✐❞❡❞ ✐♥t♦ t✇♦ ♣❛rts ✐♥ ✇❤✐❝❤ t❤❡② ❝♦♠❜✐♥❛✲ t✐♦♥ ♠♦✈❡♠❡♥ts t♦❣❡t❤❡r✿ ❛ ♣❛rt ✐♥❝❧✉❞❡s ✺ ❞❡✲ ❣r❡❡s ♦❢ ❢r❡❡❞♦♠ r♦❜♦t ❛r♠ ❛♥❞ ❛ ♣❛rt ❝r❡❛t❡❞ ❜② t❤❡ r♦t❛t✐♥❣ ♠♦t✐♦♥ ♦❢ ❛ ♣✐♣❡✳ P❛r❛♠❡t❡rs ♦❢ t❤❡ ♠♦❞❡❧ ❛r❡ ❣✐✈❡♥ ✐♥ ❚❛❜✳ ✶✳ ❇❛s❡❞ ♦♥ t❤❡ ❉❡♥❛✈✐t✲❍❛rt❡♥❜❡r❣ ❝♦♥✈❡♥✲ t✐♦♥✱ ✇❡ ✜♥❞ t❤❡s❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛tr✐❝❡s ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ❚❛❜✳ ✶✿ i ❉❡♥❛✈✐t✲❍❛rt❡♥❜❡r❣ ♣❛r❛♠❡t❡rs ♦❢ ❘♦❜♦t✳ αi ❞❡❣ 90 90 0 90 0 ♠♠ 0 157 120 50 di ♠♠ 260 371 0 210 θi ❞❡❣ θ0 θ1 θ2 θ3 θ4 θ5 ✸✳✷✳ ❖♣❡r❛t✐♥❣ r❛♥❣❡ ❞❡❣ −180 t♦ 180 −140 t♦ 140 −45 t♦ 45 −90 t♦ 90 −145 t♦ 90 −180 t♦ 180 0 −1 T, T, T, T, T ❛♥❞ T ✳ ❚❤❡ tr❛♥s❢♦r♠❛✲ t✐♦♥ ♠❛tr✐① −1 T ❞❡s❝r✐❜❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♣✐♣❡ ✐♥ ❣❧♦❜❛❧ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0}✱ ❣✐✈❡♥ ❜② ❊q✳ ✭✷✸✮✿ 0 −1 d0 −c0 s0 0 ✭✷✸✮ −1 T = s c0 0 0 0 | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r ■♥✈❡rs❡ ❑✐♥❡♠❛t✐❝s ■♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s r❡s✉❧ts ✐♥ ❡①❛❝t ♣♦s✐t✐♦♥s ♦❢ ❥♦✐♥ts ✇❤❡♥ ♣♦s✐t✐♦♥ ❛♥❞ ❞✐r❡❝t✐♦♥ ♦❢ ❡♥❞✲❡✛❡❝t♦r ✐s ❦♥♦✇♥✳ ❊q✉❛t✐♥❣ ❡♥tr✐❡s ✭✶✱✹✮ ❛♥❞ ✭✷✱✹✮ ✐♥ t❤❡ ♠❛tr✐① ✭❊q✳ ✭✷✹✮✮✱ ✇❡ ❤❛✈❡ tan θ1 ✭❬✷❪ ♦r ❬✻❪✮ ❛s ❊q✳ ✭✸✼✮✿ py = tanθ1 px ✭✸✼✮ ❲❡ ♦❜t❛✐♥ t❤❡ ✜rst ❛♥❣❧❡ ❛s ❊q✳ ✭✸✽✮✳ θ1 = arctan2(py , px ) ✭✸✽✮ ❇② ♠✉❧t✐♣❧②✐♥❣ s1 ❛♥❞ c1 ✇✐t❤ ❡❧❡♠❡♥ts ✭✶✱✶✮✱ ✭✷✱✶✮✱ ✭✶✱✷✮ ❛♥❞ ✭✷✱✷✮ ✐♥ ❊q✳ ✭✷✹✮ t❤❡♥ s❤♦rt❡♥ t❤❡♠✱ ✇❡ ✜♥❞ tan θ5 ❛s ❊q ✭✸✾✮✿ s1 nx − c1 ny = tan θ5 s1 ox − c1 oy ✭✸✾✮ ▼❛tr✐① ❞❡s❝r✐❜❡s t❤❡ ❡♥❞✲❡✛❡❝t♦r ❞✐r❡❝t✐♦♥ ❛♥❞ ♣♦s✐t✐♦♥ ✐♥ t❤❡ ❝♦♦r❞✐♥❛t❡ ❢r❛♠❡ {0} ✭s❡❡ ❬✷❪ ■♥❢❡r θ5 ❛s ❊q ✭✹✵✮✿ ♦r ❬✺❪✮ ✐s ❣✐✈❡♥ ❜② ❊q✳ ✭✷✹✮✿ θ5 = arctan2(s1 nx − c1 ny , s1 ox − c1 oy ) ✭✹✵✮ nx ax ox px ny ay oy py 0 , ❚♦ ✜♥❞ θ3 ✱ ✇❡ ♥❡❡❞ t♦ ✜♥❞ θ234 = θ2 + θ3 + T =1 T T T T T = n az oz pz z θ4 ✳ ❊q✉❛t✐♥❣ ❡♥tr✐❡s ✭✶✱✸✮✱ ✭✷✱✸✮ ❛♥❞ ✭✸✱✸✮ ✐♥ t❤❡ 0 ♠❛tr✐① ❊q✳ ✭✷✹✮✱ ✇❡ ♦❜t❛✐♥ ❊q ✭✹✶✮✿ ✭✷✹✮ c1 ax + s1 ay = tan θ234 ✭✹✶✮ ✇❤❡r❡ −az nx = c1 c234 c5 + s1 s5 , ✭✷✺✮ ■♥❢❡r θ234 ❛s ❊q ✭✹✷✮✿ n =s c c −c s , ✭✷✻✮ 5T y 234 5 nz = s234 c5 , ✭✷✼✮ ax = −c1 c234 s5 + s1 c5 , ✭✷✽✮ ay = −s1 c234 s5 − c1 c5 , ✭✷✾✮ az = −s234 s5 , ✭✸✵✮ ox = c1 s234 , ✭✸✶✮ oy = s1 s234 , ✭✸✷✮ oz = −c234 , ✭✸✸✮ px = d5 c1 s234 + a4 c1 c234 + a3 c1 c23 + a2 c1 c2 , ✭✸✹✮ py = d5 s1 s234 + a4 s1 c234 + a3 s1 c23 + a2 s1 c2 , ✭✸✺✮ pz = −d5 c234 + a4 s234 + a3 s23 + a2 s2 + d1 , ✭✸✻✮ θ234 = arctan2(c1 ax + s1 ay , −az ) ✭✹✷✮ ❆❢t❡r ✇❡ ❤❛✈❡ θ1 ❛♥❞ θ5 ✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛tr✐① 01 T ❛♥❞ 45 T ✳ ❇❛s❡❞ ♦♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛tr✐① 01 T ✱ ✇❡ ✜♥❞ 14 T ❛s ❊q✳ ✭✹✸✮✿ 4T = −1 −1 1T 5T 5T c234 s234 = 0 0 s234 −c234 0 a4 c234 + a3 c23 + a2 c2 a4 s234 + a3 s23 + a2 s2 ✭✹✸✮ ❲❡ s❡t✿ si ✿ sin θi ✱ ci ✿ cos θi ✱ si k ✿ sin(θi + · · · + θk )✱ ci k ✿ cos(θi + · · · + θk )✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ x41 = a4 c234 + a3 c23 + a2 c2 , y41 = a4 s234 + a3 s23 + a2 s2 ✭✹✹✮ ✾✾ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r ❘❡❛rr❛♥❣✐♥❣ t❤❡ t✇♦ ❡q✉❛t✐♦♥s ✐♥ ❊q✳ ✭✹✹✮✱ ■♥ ❈❛rt❡s✐❛♥ s♣❛❝❡✱ ❝✉tt✐♥❣ ♣❛t❤ ✇✐❧❧ ❜❡ ❞✐✲ sq✉❛r✐♥❣ t❤❡♠ ❛♥❞ t❤❡♥ ❛❞❞✐♥❣ t❤❡ sq✉❛r❡s ❣✐✈❡s ✈✐❞❡❞ ✐♥t♦ s❡t ♦❢ ♣♦✐♥ts ✐♥ ✇❤✐❝❤ t❤❡ s♣❛❝❡ ❜❡✲ ❊q✳ ✭✹✺✮ ❛♥❞ ❊q✳ ✭✹✻✮✿ t✇❡❡♥ t❤❡s❡ ♣♦✐♥ts ✐s ✈❡r② s♠❛❧❧ ❛♥❞ ❡q✉❛❧ ∆p✳ ■♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s ✐s ✉s❡❞ t♦ ❞❡✜♥❡ ❥♦✐♥t ✈❛r✐✲ (x41 − a4 c234 )2 + (y41 − a4 s234 )2 − a23 − a22 ❛❜❧❡s ✐♥ ❥♦✐♥t s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s❡t ♦❢ ♣♦✐♥ts c3 = , 2a3 a2 ✐♥ ❈❛rt❡s✐❛♥ s♣❛❝❡ ✭❬✽❪✱ ❬✾❪ ♦r ❬✶✵❪✮✳ ✭✹✺✮ ❏♦✐♥t ♣❛t❤ ♣❧❛♥♥✐♥❣ ♠✉st ❡♥s✉r❡ t❤❡ ❝♦♥t✐♥✉✲ s3 = ± − c23 ✭✹✻✮ ✐t② ♦❢ ♣♦s✐t✐♦♥✱ ✈❡❧♦❝✐t②✱ ❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ s✉✐t❛❜❧❡ ❝❤♦✐❝❡✳ ■♥❢❡r θ3 ❛s ❊q ✭✹✼✮✿ ❙✉♣♣♦s❡ t❤❛t nth ❥♦✐♥t r♦t❛t❡s ❛♥❣❧❡s θi ❛♥❞ θ3 = arctan2(s3 , c3 ) ✭✹✼✮ θi+1 ✐♥ ❞✉r❛t✐♦♥s tk ❛♥❞ tk+1 r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠ ♦❢ nth ❥♦✐♥t✿ ❙❡t✿ θi (t) = ai3 t3 + ai2 t2 + ai1 t + ai0 k1 = a3 c3 + a2 , k2 = a3 s3 ✭✹✽✮ x41 − a4 c234 = k1 c2 − k2 s2 , y41 − a4 s234 = k1 s2 + k2 c2 ❱❡❧♦❝✐t②✿ θ˙i (t) = 3ai3 t2 + 2ai2 t + ai1 ❋r♦♠ ❊q✳ ✭✹✹✮ ❛♥❞ ❊q✳ ✭✹✽✮✱ ✇❡ ♦❜t❛✐♥✿ ✭✹✾✮ ❙❡t✿ P❛r❛♠❡t❡rs ♦❢ t❤❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ♠✉st ❜❡ ❞❡✜♥❡❞ ❜❛s❡❞ ♦♥ ❝♦♥str❛✐♥ts s♦ t❤❛t ❥♦✐♥t ♣❛t❤ s❛t✐s❢② t❤❡ ❝♦♥t✐♥✉✐t② ♦❢ ♣♦s✐t✐♦♥✱ ✈❡❧♦❝✐t② ❛♥❞ ❛❝❝❡❧❡r❛t✐♦♥✳ ✭✺✵✮ ❇❛s❡❞ ♦♥ ❊q✳ ✭✹✾✮ ❛♥❞ ❊q✳ ✭✺✵✮✱ ✇❡ ♦❜t❛✐♥ ❊q✳ ✭✺✶✮✳ x − a4 c234 cos(γ + θ2 ) = 41 , k3 ✭✺✶✮ y − a s 41 234 sin(γ + θ2 ) = k3 ❋r♦♠ ❊q✳ ✭✺✶✮✱ ✇❡ ♦❜t❛✐♥ θ2 ✭❬✷❪ ♦r ❬✼❪✮ ❛s ❊q✳ ✭✺✷✮✿ − arctan2(k2 , k1 ) ✭✺✷✮ ❆♥❞ ✇❡ ❤❛✈❡ θ4 ❛s ❊q✳ ✭✺✸✮✿ θ4 = θ234 − θ3 − θ2 ✭✺✸✮ ❚r❛❥❡❝t♦r② P❧❛♥♥✐♥❣ ▲✐♥❦s ✇✐❧❧ ❜❡ ♠♦✈❡❞ ❝♦♥❝✉rr❡♥t❧② ❛♥❞ ❝♦r♣♦✲ r❛t❡❧② s♦ t❤❛t ❡♥❞✲❡✛❡❝t♦r ✇✐❧❧ ❢♦❧❧♦✇ ❝✉tt✐♥❣ ❡q✉❛t✐♦♥ ✐♥ ❞❡✜♥❡❞ ❞✉r❛t✐♦♥ t✳ θi (0) = θk−1 , ✭✺✻✮ θi (tk ) = θi+1 (0) = θk , ✭✺✼✮ θi+1 (tk+1 ) = θk+1 ✭✺✽✮ ❱❡❧♦❝✐t② ❝♦♥str❛✐♥ts✿ θ˙i (0) = θ˙k−1 , θ˙i (tk ) = θ˙i+1 (0) = θ˙k , ✭✺✾✮ θ˙i+1 (tk+1 ) = θ˙k+1 rt strts ăi (tk ) = ăi+1 (0) = ăk = arctan2(y41 a4 s234 , x41 − a4 c234 ) ✶✵✵ ✭✺✺✮ P♦s✐t✐♦♥ ❝♦♥str❛✐♥ts✿ k3 = k12 + k22 , γ = arctan2(k2 , k1 ) ✸✳✸✳ ✭✺✹✮ ✭✻✷✮ ❚❤❡ ❞✉r❛t✐♦♥s ❛r❡ ❡q✉❛❧✿ tk = tk+1 = ∆t✱ s♦❧✈✐♥❣ ❊q✳ ✭✺✻✮✱ ❊q✳ ✭✺✼✮✱ ❊q✳ ✭✺✽✮✱ ❊q✳ ✭✺✾✮✱ ❊q✳ ✭✻✵✮✱ ❊q✳ ✭✻✶✮ ❛♥❞ ❊q✳ ✭✻✷✮✱ ✇❡ ♦❜t❛✐♥✿ −2(θk − θk−1 ) + (θ˙k + θ˙k−1 )∆t , ∆t3 6(θk − θk−1 ) − 2(θ˙k + 2θ˙k−1 )∆t = , 2∆t2 = θ˙k−1 , ai3 = ✭✻✸✮ ai2 ✭✻✹✮ ai1 ai0 = θk−1 ✭✻✺✮ ✭✻✻✮ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r P❧✉❣❣✐♥❣ ❊q✳ ✭✻✸✮✱ ❊q✳ ✭✻✹✮✱ ❊q✳ ✭✻✺✮ ❛♥❞ ✭✻✻✮ ✐♥t♦ ❊q✳ ✭✺✹✮ ❛♥❞ ❊q✳ ✭✺✺✮✱ ✇❡ ✜♥❞ t❤❛t✿ −2(θk − θk−1 ) + (θ˙k + θ˙k−1 )∆t t ∆t3 6(θk − θk−1 ) − 2(θ˙k + 2θ˙k−1 )∆t + t 2∆t2 + θ˙k−1 t + θk−1 , ✭✻✼✮ ˙ ˙ −2(θk − θk−1 ) + (θk + θk−1 )∆t t θ˙i (t) = ∆t3 6(θk − θk−1 ) − 2(θ˙k + 2θ˙k−1 )∆t +2 t 2∆t2 + θ˙k−1 ✭✻✽✮ θi (t) = ✹✳ ❋✐❣✳ ✻✿ ❚❤❡ ✸❉ s②st❡♠ ♦❢ ❘♦❜♦t✳ ❋✐❣✳ ✼✿ ❚❤❡ ❘♦❜♦t s②st❡♠ ✐♥ ▼❛t❧❛❜ ❙✐♠✉❧✐♥❦✳ ❙✐♠✉❧❛t✐♦♥ ❛♥❞ ❈♦♥tr♦❧ ❚r❛❥❡❝t♦r② ❛♥❞ ❞✐r❡❝t✐♦♥ ✇❡r❡ ♣❧❛♥♥❡❞ ✐♥ ❈❛rt❡✲ s✐❛♥ s♣❛❝❡✳ ❚❤❡② ✇✐❧❧ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ t❤❡ ❥♦✐♥t s♣❛❝❡✳ ❚❤❡ r♦❜♦t ✐s ❝♦♥tr♦❧❧❡❞ ✐♥ t❤❡ ❥♦✐♥t s♣❛❝❡ s♦ t❤❛t t❤❡ ❡♥❞✲❡✛❡❝t♦r ❢♦❧❧♦✇s t❤❡ tr❛❥❡❝✲ t♦r② ❛♥❞ t❤❡ ❡①♣❡❝t❡❞ ❞✐r❡❝t✐♦♥✳ ❈♦♥tr♦❧ ✢♦✇ ❝❤❛rt ❛❧❣♦r✐t❤♠ ✐s ❣✐✈❡♥ ❜② ❋✐❣✳ ✺✳ ✇❤❡r❡ • V ✿ ❚❤❡ ♦✉t♣✉t ♦❢ t❤❡ ❝♦♥tr♦❧❧❡r✱ • E ✿ ❊rr♦r ❜❡t✇❡❡♥ t❤❡ ✐♥♣✉t ✈❛❧✉❡ ❛♥❞ t❤❡ ❢❡❡❞❜❛❝❦ ✈❛❧✉❡✱ • Kp , Kd , Ki ✿ Pr♦♣♦rt✐♦♥❛❧ ❣❛✐♥✱ ❞❡r✐✈❛t✐✈❡ ❣❛✐♥✱ ✐♥t❡❣r❛❧ ❣❛✐♥ ❛♥❞ ❋✐❣✳ ✺✿ ❈♦♥tr♦❧ ✢♦✇❝❤❛rt ❛❧❣♦r✐t❤♠✳ • τf ✿ ❚❤❡ t✐♠❡ ❝♦♥st❛♥t ♦❢ t❤❡ ✜rst ♦r❞❡r ✜❧✲ t❡r✳ ❚❤❡ r♦❜♦t ✐s ❞r❛✇♥ ❜② ❙♦❧✐❞❲♦r❦s ❛s ❋✐❣✳ ✻✳ ❚❤✐s ❘♦❜♦t s②st❡♠ ✐s ✐♠♣♦rt❡❞ ✐♥t♦ ▼❛t❧❛❜ ❙✐♠✉❧✐♥❦ ❛s ❋✐❣✳ ✼✳ Kd tf P■❉ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ✜rst ♦r❞❡r ✜❧t❡r ✭❬✶✶❪ ❛♥❞ ❬✶✷❪✮ ✐s ❣✐✈❡♥ ❜② ❊q✳ ✭✻✾✮✿ Kd Kp Ki Ki s + s+ K s + K + d p V (s) τf τf s = τf = , s E(s) τf s + s + τf ✭✻✾✮ V Kp tf E r f x&1 s x1 x&2 s x2 Ki tf tf ❋✐❣✳ ✽✿ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ P■❉ ❝♦♥tr♦❧❧❡r ✇✐t❤ ✜rst ♦r❞❡r ✜❧t❡r✳ ✶✵✶ ❋r♦♠ ❋✐❣✳ ✽✱ ✇❡ ❤❛✈❡✿ x˙ = − x1 + e, τf ✭✼✵✮ x˙ = x1 , Ki Kp Kd V = x2 + x1 + x˙ τf τf τf ✭✼✶✮ Kd Kp − τf τf x1 + Ki Kd x2 + e τf τf ✭✼✷✮ -100 -200 -300 ref0 ref1 ❋✐❣✳ ✾✿ ref2 ref3 ref5 resp0 resp1 15 resp2 resp3 20 resp4 25 resp5 30 ❚r❛❥❡❝t♦r② ❛♥❞ r❡s♣♦♥s❡ ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❝❛s❡ ♦❢ st❛t✐❝ ♣✐♣❡✳ -100 -200 -300 ref0 ref1 ref2 ref3 ref4 10 ref5 resp0 resp1 15 resp2 resp3 20 resp4 25 resp5 30 time (s) ❋✐❣✳ ✶✵✿ ❚r❛❥❡❝t♦r② ❛♥❞ r❡s♣♦♥s❡ ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❝❛s❡ ♦❢ r♦t❛r② ♣✐♣❡✳ x˙ X˙ = , x˙ Kd τf2 ; B= , ✭✼✻✮ 0.05 -0.05 error0 -0.1 Ki ; τf Kd D= τf ✭✼✼✮ ❋✐❣✳ ✶✶✿ X˙ = AX + Be, ✭✼✽✮ V = CX + De ✭✼✾✮ error2 error3 15 error4 error5 20 25 30 ❏♦✐♥t ❡rr♦rs ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❝❛s❡ ♦❢ st❛t✐❝ ♣✐♣❡✳ 0.1 0.05 -0.05 error0 -0.1 ❚❤❡ s✐♠✉❧❛t❡❞ r❡s✉❧ts✿ error1 10 time (s) joints errors (deg) 0 ✭✼✺✮ joints errors (deg) 0.1 ❋r♦♠ ❊q✳ ✭✼✺✮✱ ❊q✳ ✭✼✻✮ ❛♥❞ ❊q✳ ✭✼✼✮✱ ✇❡ ✜♥❞ t❤❛t✿ error1 10 error2 error3 15 error4 20 error5 25 30 time (s) ❚❤❡ s✐♠✉❧❛t✐♦♥ t✐♠❡ ✐s t❤❡ 30 s✳ ❋✐❣✉r❡✳ ✾ ❛♥❞ ❋✐❣✳ ✶✵ ❣✐✈❡ tr❛❥❡❝t♦r② ❛♥❞ r❡s♣♦♥s❡ ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❥♦✐♥t s♣❛❝❡✳ ❋✐❣✉r❡ ✶✶ ❛♥❞ ❋✐❣✳ ✶✷ ❛r❡ t❤❡ ❡rr♦r ❣r❛♣❤s ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❥♦✐♥t s♣❛❝❡✳ ■♥ ❋✐❣✳ ✶✶✿ ❙✐♥❝❡ t❤❡ st❛t✐❝ ♣✐♣❡ ❧❡❛❞s t♦ t❤❡ ❥♦✐♥t ✐s ♠♦t✐♦♥❧❡ss ❛♥❞ ❡rr♦r❧❡ss✱ t❤❡ ♠❛①✐♠✉♠ ❡rr♦r ❜❡❧♦♥❣s t♦ ❥♦✐♥ts 3✱ ❛♥❞ ✇✐t❤ ❛ ♠❛①✐♠✉♠ ✈❛❧✉❡ ♦❢ 0.04◦ ✳ ❚❤❡ ❜❡st ❛❝t✐✈✐t② ✐s ❥♦✐♥t ✇✐t❤ ❛ ♠❛①✐♠✉♠ ❡rr♦r ♦❢ 0.0075◦ ✳ ■♥ ❋✐❣✳ ✶✷✿ ❙✐♥❝❡ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ♦♥ t❤❡ str❛✐❣❤t s❡❝t✐♦♥✱ ❥♦✐♥t st❛②s st✐❧❧✳ ✶✵✷ ref4 10 100 -400 ❙❡t✿ x1 ; x2 − A= τf Kp − C= τf time (s) ■♥ st❛t❡ s♣❛❝❡✱ ❊q✳ ✭✼✵✮✱ ❊q✳ ✭✼✶✮ ❛♥❞ ❊q✳ ✭✼✷✮ ❛r❡ ❣✐✈❡♥ ❜② ❊q✳ ✭✼✸✮ ❛♥❞ ❊q✳ ✭✼✹✮✿ − x˙ x1 + e, = τf ✭✼✸✮ x˙ x2 Kd Ki x1 Kp Kd − e ✭✼✹✮ + V = τf τf τf x2 τf X= | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r 100 -400 responses of joints (deg) = responses of joints (deg) ❱❖▲❯▼❊✿ ✶ ❋✐❣✳ ✶✷✿ ❏♦✐♥t ❡rr♦rs ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❝❛s❡ ♦❢ r♦t❛r② ♣✐♣❡✳ ❚❤❡ ♠❛①✐♠✉♠ ❡rr♦r ❜❡❧♦♥❣s t♦ ❥♦✐♥t ✇✐t❤ ❛ ✈❛❧✉❡ ♦❢ 0.009◦ ✳ ❚❤❡ ❜❡st ❛❝t✐✈✐t② ✐s ❥♦✐♥t ✇✐t❤ ❛ ♠❛①✐♠✉♠ ❡rr♦r ♦❢ 0.0006◦ ✳ ❋✐❣✉r❡ ✶✸ ❛♥❞ ❋✐❣✳ ✶✹ ❛r❡ t❤❡ ❡rr♦r ❣r❛♣❤s ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ✐♥ t❤❡ t❤r❡❡ ❛①❡s ♦❢ X −Y −Z ✐♥ t❤❡ ❈❛rt❡s✐❛♥ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✇♦ ❝❛s❡s✿ st❛♥❞✐♥❣ ❛♥❞ r♦t❛t✐♥❣✳ ■♥ ❋✐❣✳ ✶✸✿ P♦s✐t✐♦♥ ❡rr♦r ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r position errors (mm) 0.2 0.1 -0.1 X -0.2 10 Y Z 15 20 25 30 time (s) ❋✐❣✳ ✶✸✿ P♦s✐t✐♦♥ ❡rr♦rs ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ✐♥ t❤❡ ❝❛s❡ ♦❢ st❛t✐❝ ♣✐♣❡✳ 0.1 ❋✐❣✳ ✶✺✿ ❊①♣❡r✐♠❡♥t❛❧ ❘♦❜♦t s②st❡♠✳ 0.1 -0.1 X -0.2 10 Y 15 Z 20 25 30 time (s) ❋✐❣✳ ✶✹✿ P♦s✐t✐♦♥ ❡rr♦rs ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ✐♥ t❤❡ ❝❛s❡ ♦❢ r♦t❛r② ♣✐♣❡✳ joints errors (deg) position errors (mm) 0.2 0.05 -0.05 joint -0.1 joint joint joint 10 joint joint 15 20 25 30 time (s) ❉✐s❝✉ss✐♦♥✿ ❚❤❡ s✐♠✉❧❛t❡❞ r❡s✉❧ts s❤♦✇❡❞ t❤❛t t❤❡ ❝❛s❡ ♦❢ ❛ st❛t✐❝ ♣✐♣❡ ❝✉tt✐♥❣ ✇❛s ♥♦t ❛s ❣♦♦❞ ❛s t❤❡ ❝❛s❡ ♦❢ ❛ r♦t❛r② ♣✐♣❡ ❝✉tt✐♥❣✳ ✺✳ ❊①♣❡r✐♠❡♥t ❋✐❣✳ ✶✻✿ ❏♦✐♥t ❡rr♦rs ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❝❛s❡ ♦❢ st❛t✐❝ ♣✐♣❡✳ 0.1 joints errors (deg) r❛♥❣❡s ❢r♦♠ −0.022 t♦ 0.026 ♠♠✳ ■♥ ❋✐❣✳ ✶✹✿ P♦s✐t✐♦♥ ❡rr♦r r❛♥❣❡s ❢r♦♠ −0.021 t♦ 0.020 ♠♠✳ 0.05 -0.05 joint -0.1 joint joint joint 10 joint joint 15 20 25 30 time (s) ❋✐❣✉r❡ ✶✺ ✐s t❤❡ r❡❛❧ r♦❜♦t s②st❡♠✳ ❚✐♠❡ t♦ ✜♥✲ ❋✐❣✳ ✶✼✿ ❏♦✐♥t ❡rr♦rs ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❝❛s❡ ♦❢ r♦t❛r② ♣✐♣❡✳ ✐s❤✐♥❣ ✇♦r❦ ♦❢ ❘♦❜♦t ✐s s❡t t♦ 30 s✳ ❚✇♦ ♠✐❝r♦✲ ❝♦♥tr♦❧❧❡rs ✇✐❧❧ ❝♦♥tr♦❧ ✜✈❡ ❤❛r♠♦♥✐❝ ❞r✐✈❡r ♠♦✲ t♦rs ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✜✈❡ ❥♦✐♥ts ♦❢ ❘♦❜♦t ❛♥❞ ❛ st❛♥❞✐♥❣ ❛♥❞ r♦t❛t✐♥❣✳ ❲❡ s❡❡ t❤❛t t❤❡ ❡rr♦rs r♦t❛r② ♠♦t♦r✳ ❉❛t❛ ♦❜t❛✐♥❡❞ ❢r♦♠ ❘♦❜♦t ✇✐❧❧ ❜❡ ✐♥ t❤❡ t✇♦ ❣r❛♣❤s r❛♥❣❡ ❢r♦♠ −0.2 t♦ 0.2 ♠♠✳ tr❛♥s♠✐tt❡❞ t♦ t❤❡ ❝♦♠♣✉t❡r✳ ❋✐❣✳ ✶✽ ❤❛s ❛ ❧❛r❣❡r ❡rr♦r ❣r❛♣❤ ❛♥❞ ✐s ♠♦r❡ ♦s✲ ❝✐❧❧❛t✐♥❣ t❤❛♥ ❋✐❣✳ ✶✾✳ ❚❤❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✿ ❉✐s❝✉ss✐♦♥✿ ❚❤❡ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts s❤♦✇❡❞ ❋✐❣✉r❡ ✶✻ ❛♥❞ ❋✐❣✳ ✶✼ ❛r❡ t❤❡ ❡rr♦r ❣r❛♣❤s ♦❢ s✐① ❥♦✐♥ts ✐♥ t❤❡ ❥♦✐♥t s♣❛❝❡✳ ■♥ ❋✐❣✳ ✶✻✿ ❚❤❡ t❤❛t t❤❡ ❝❛s❡ ♦❢ ❛ r♦t❛r② ♣✐♣❡ ❝✉tt✐♥❣ ✇❛s ❜❡tt❡r ♠❛①✐♠✉♠ ❡rr♦r ❜❡❧♦♥❣s t♦ ❥♦✐♥ts ❛♥❞ 4❀ t❤❡ t❤❛♥ t❤❡ ❝❛s❡ ♦❢ ❛ st❛t✐❝ ♣✐♣❡ ❝✉tt✐♥❣✱ ✇✐t❤ ❧❡ss ❡rr♦r r❛♥❣❡s ❢r♦♠ −0.072◦ t♦ 0.079◦ ✳ ■♥ ❋✐❣✳ ✶✼✿ ❡rr♦r ❛♥❞ ❧❡ss ♦s❝✐❧❧❛t✐♦♥ ❡rr♦r✳ ❚❤❡ ♠❛①✐♠✉♠ ❡rr♦r ❜❡❧♦♥❣s t♦ ❥♦✐♥ts ❛♥❞ 4❀ t❤❡ ❡rr♦r r❛♥❣❡s ❢r♦♠ −0.058◦ t♦ 0.045◦ ✳ ❚❤❡s❡ ❡rr♦rs ❛r❡ s♠❛❧❧❡r ✐♥ ❋✐❣✳ ✶✻✳ ✻✳ ❈♦♥❝❧✉s✐♦♥ ❋✐❣✉r❡ ✶✽ ❛♥❞ ❋✐❣✳ ✶✾ ❛r❡ t❤❡ ❡rr♦r ❣r❛♣❤s ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ✐♥ t❤❡ t❤r❡❡ ❛①❡s ♦❢ X −Y −Z ✐♥ ❚❤❡ ♣❛♣❡r ❤❛s s♦❧✈❡❞ t❤❡ ✇❤♦❧❡ ♣r♦❜❧❡♠✿ ❜✉✐❧❞✲ t❤❡ ❈❛rt❡s✐❛♥ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t✇♦ ❝❛s❡s✿ ✐♥❣ t❤❡ ❝✉tt✐♥❣ tr❛❥❡❝t♦r②✱ s♦❧✈✐♥❣ t❤❡ ✐♥✈❡rs❡ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✵✸ ❱❖▲❯▼❊✿ ✶ position errors (mm) 0.3 ❘❡❢❡r❡♥❝❡s 0.2 0.1 -0.1 -0.2 X -0.3 Y Z 10 15 20 25 30 time (s) ❋✐❣✳ ✶✽✿ P♦s✐t✐♦♥ ❡rr♦rs ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ✐♥ t❤❡ ❝❛s❡ ♦❢ st❛t✐❝ ♣✐♣❡✳ 0.3 position errors (mm) | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r 0.2 0.1 ❬✶❪ ▼❖❍❆❘❆◆❆✱ ❇✳✱ ❘✳ ●❯P❚❆ ❛♥❞ ❇✳ ❑✳ ❑❯❙❍❲❆❍❆✳ ❖♣t✐♠✐③❛t✐♦♥ ❛♥❞ ❉❡s✐❣♥ ♦❢ ❛ ▲❛s❡r✲❈✉tt✐♥❣ ▼❛❝❤✐♥❡ ✉s✐♥❣ ❉❡❧t❛ ❘♦❜♦t✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❊♥❣✐♥❡❡r✐♥❣ ❚r❡♥❞s ❛♥❞ ❚❡❝❤♥♦❧♦❣②✳ ✷✵✶✹✱ ✈♦❧✳ ✶✵✱ ♥♦✳ ✹✱ ♣♣✳ ✶✼✻✕✶✼✾✳ ❬✷❪ ◗❯❖❈ ❇❆❖✱ ❉✳✱ ❚✳ ❚✳ P❍❆◆ ❛♥❞ ❚✳ ❚✳ ◆●❯❨❊◆✳ ❙t✉❞② ♦♥ ❈♦♥tr♦❧ ♦❢ P✐♣❡ ❈✉tt✐♥❣ ❘♦❜♦t✳ ■♥✿ ■♥t❡r♥❛t✐♦♥❛❧ ❙②♠♣♦✲ s✐✉♠ ♦♥ ▼❡❝❤❛tr♦♥✐❝s ❛♥❞ ❘♦❜♦t✐❝s✳ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✿ ❍❈▼❯❚✱ ✷✵✶✸✱ ♣♣✳ ✶✶✽✕✶✷✸✳ -0.1 -0.2 X -0.3 Y Z 10 15 20 25 30 time (s) ❋✐❣✳ ✶✾✿ P♦s✐t✐♦♥ ❡rr♦rs ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ✐♥ t❤❡ ❝❛s❡ ♦❢ r♦t❛r② ♣✐♣❡✳ ❦✐♥❡♠❛t✐❝s✱ ♣❧❛♥♥✐♥❣ t❤❡ tr❛❥❡❝t♦r② ♦❢ ♠♦t✐♦♥✱ s✐♠✉❧❛t✐♥❣ ❛♥❞ ❝♦♥tr♦❧❧✐♥❣ ❘♦❜♦t ✐♥ r❡❛❧✐t②✳ ▼♦r❡ ✐♠♣♦rt❛♥t❧②✱ t❤✐s ♣❛♣❡r ❤❛s ❞❡✈❡❧♦♣❡❞ t✇♦ ❞✐✛❡r❡♥t ♣✐♣❡ ❝✉tt✐♥❣ s♦❧✉t✐♦♥s✱ ❛♥❞ ❣✐✈❡s t❤❡ ❝♦♠♣❛r❛t✐✈❡ r❡s✉❧ts ❜❡t✇❡❡♥ t❤❡ t✇♦ ♦♥❡s ✐♥ ❜♦t❤ s✐♠✉❧❛t✐♦♥ ❛♥❞ ❡①♣❡r✐♠❡♥t✳ ❚❤❡s❡ ❝♦♠✲ ♣❛r❛t✐✈❡ r❡s✉❧ts s❤♦✇ t❤❛t ♠❡t❤♦❞ ♦❢ t❤❡ ❡♥❞✲ ❡✛❡❝t♦r ♠♦✈❡s ♦♥ ❛ str❛✐❣❤t ❧✐♥❡ ✇❤✐❧❡ t❤❡ r♦t❛t✲ ✐♥❣ ♣✐♣❡ t♦ ❝r❡❛t❡ t❤❡ ❝✉tt✐♥❣ ♣❛t❤ ❛♥❞ ❞✐r❡❝t✐♦♥ ❢♦r ❜❡tt❡r t❤❛♥ ♠❡t❤♦❞ ♦❢ t❤❡ ❡♥❞✲❡✛❡❝t♦r ♠♦✈❡s ♦♥ ❝✉tt✐♥❣ ♣❛t❤ ❛♥❞ ❞✐r❡❝t✐♦♥ ✇❤✐❧❡ t❤❡ st❛t✐♦♥✲ ❛r② ♣✐♣❡✳ ❚❤✐s ❝♦♥❝❧✉s✐♦♥ ✐s ❛♥ ✐♠♣♦rt❛♥t ♥♦t❡ t❤❛t ✇❡ s❤♦✉❧❞ ❞❡s✐❣♥ t❤❡ r♦❜♦t ❛r♠ ❛♥❞ t❤❡ ♣✐♣❡ ❝♦♦r❞✐♥❛t❡ ♠♦✈❡♠❡♥t t♦❣❡t❤❡r✱ ❜r✐♥❣ t❤❡ ❜❡st ❡✛❡❝t✳ ❆❝❦♥♦✇❧❡❞❣♠❡♥t P✐♣❡❧✐♥❡ P❧❛♥♥✐♥❣ ❛♥❞ ❈♦♥str✉❝t✐♦♥ ❋✐❡❧❞ ▼❛♥✉❛❧✳ ❬✹❪ ❈❤❛♣t❡r ✶✼✳ ▼❊◆❖◆✱ ❊✳ ❙✳ ❲❛❧t❤❛♠✿ ●✉❧❢ Pr♦❢❡ss✐♦♥❛❧ P✉❜❧✐s❤✐♥❣✱ ✷✵✶✶✱ ♣♣✳ ✸✺✼✕✸✼✽✳ ❬✺❪ ❱❊❘▼❆✱ ❆✳ ❛♥❞ ❱✳ ❆✳ ❉❊❙❍P❆◆❉❊✳ ❊♥❞✲❡✛❡❝t♦r ♣♦s✐t✐♦♥ ❛♥❛❧②s✐s ♦❢ ❙❈❖❘❇❖❚✲❊❘✲❱♣❧✉s ❘♦❜♦t✳ ■♥t❡r♥❛✲ t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❙♠❛rt ❍♦♠❡✳ ✷✵✶✶✱ ✈♦❧✳ ✺✱ ♥♦✳ ✶✱ ♣♣✳ ✶✕✻✳ ❬✻❪ ❉❊❙❍P❆◆❉❊✱ ❱✳ ❆✳ ❛♥❞ P✳ ▼✳ ●❊❖❘●❊✳ ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ ❢♦r ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s ♦❢ ❙❈❖❘❇❖❚✲❊❘✲❱♣❧✉s ❘♦❜♦t✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❊♠❡r❣✐♥❣ ❚❡❝❤♥♦❧♦❣② ❛♥❞ ❆❞✲ ✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣✳ ✷✵✶✷✱ ✈♦❧✳ ✷✱ ✐ss✳ ✸✱ ♣♣✳ ✹✼✽✕✹✽✶✳ ❬✼❪ ❳❯✱ ❉✳✱ ❈✳ ❆✳ ❆❈❖❙❚❆ ❈❆▲❉❊❘❖◆✱ ❏✳ ◗✳ ●❆◆✱ ❍✳ ❍❯ ❛♥❞ ▼✳ ❚❆◆✳ ❆♥ ❛♥❛❧②s✐s ♦❢ t❤❡ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s ❢♦r ❛ ✺✲ ❉❖❋ ♠❛♥✐♣✉❧❛t♦r✳ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❆✉t♦♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t✐♥❣✳ ✷✵✵✺✱ ✈♦❧✳ ✷✱ ✐ss✳ ✷✱ ♣♣✳ ✶✶✹✕✶✷✹✳ ❬✽❪ ❏❆❩❆❘✱ ❘✳ ◆✳ ❚❤❡ ✇♦r❦ ✇❛s s✉♣♣♦rt❡❞ ❜② Pr♦❢❡ss♦r ■✈❛♥ ❩❡❧✐♥❦❛✱ ❚♦♥ ❉✉❝ ❚❤❛♥❣ ❯♥✐✈❡rs✐t② ❛♥❞ Pr♦✲ ❢❡ss♦r ◆❣✉②❡♥ ❚❛♥ ❚✐❡♥✱ ✧❍✐✲❚❡❝❤ ▼❡❝❤❛tr♦♥✲ ✐❝s ▲❛❜♦r❛t♦r②✧ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t② ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣②✳ ✶✵✹ ❬✸❪ ❈❤❛♣t❡r ✷✳ ▼❈❆▲▲■❙❚❊❘✱ ❊✳ ❲✳ P✐♣❡❧✐♥❡ ❘✉❧❡s ♦❢ ❚❤✉♠❜ ❍❛♥❞❜♦♦❦✳ ✽t❤ ❊❞✐t✐♦♥✳ ❇♦st♦♥✿ ❊❧s❡✈✐❡r✴●✉❧❢ Pr♦❢❡ss✐♦♥❛❧ P✉❜✲ ❧✐s❤✐♥❣✱ ✷✵✶✹✱ ♣♣✳ ✻✹✕✾✶✳ ✳ ❑✐♥❡♠❛t✐❝s✱ ❚❤❡♦r② ♦❢ ❆♣♣❧✐❡❞ ❘♦❜♦t✐❝s✿ ❉②♥❛♠✐❝s✱ ❛♥❞ ❈♦♥tr♦❧✳ ✷♥❞ ❊❞✐t✐♦♥✳ ◆❡✇ ❨♦r❦✿ ❙♣r✐♥❣❡r✱ ✷✵✶✵✳ ❬✾❪ ❈❘❆■●✱ ❏✳ ❏✳ ■♥tr♦❞✉❝t✐♦♥ t♦ ❘♦❜♦t✐❝s✿ ▼❡✲ ❝❤❛♥✐❝s ❛♥❞ ❈♦♥tr♦❧✳ ✸r❞ ❊❞✐t✐♦♥✳ ❯♣♣❡r ❙❛❞❞❧❡ ❘✐✈❡r✿ P❡❛rs♦♥✴Pr❡♥t✐❝❡ ❍❛❧❧✱ ✷✵✵✺✳ ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✶ ❬✶✵❪ ❚❖◆❉❯✱ ❇✳ ❛♥❞ ❙✳ ❆✳ ❇❆❩❆❩✳ ❚❤❡ ❚❤r❡❡✲ ❈✉❜✐❝ ▼❡t❤♦❞✿ ❆♥ ❖♣t✐♠❛❧ ❖♥❧✐♥❡ ❘♦❜♦t ❏♦✐♥t ❚r❛❥❡❝t♦r② ●❡♥❡r❛t♦r ✉♥❞❡r ❱❡❧♦❝✐t②✱ ❆❝❝❡❧❡r❛t✐♦♥✱ ❛♥❞ ❲❛♥❞❡r✐♥❣ ❈♦♥str❛✐♥ts✳ ❚❤❡ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❘♦❜♦t✐❝s ❘❡✲ s❡❛r❝❤✳ ✶✾✾✾✱ ✈♦❧✳ ✶✽✱ ✐ss✳ ✾✱ ♣♣✳ ✽✾✸✕✾✵✶✳ ❬✶✶❪ ❙❑❖●❊❙❚❆❉✱ ❙✳ ❙✐♠♣❧❡ ❆♥❛❧②t✐❝ ❘✉❧❡s ❢♦r ▼♦❞❡❧ ❘❡❞✉❝t✐♦♥ ❛♥❞ P■❉ ❈♦♥tr♦❧❧❡r ❚✉♥✐♥❣✳ ❏♦✉r♥❛❧ ♦❢ Pr♦❝❡ss ❈♦♥tr♦❧✳ ✷✵✵✸✱ ✈♦❧✳ ✶✸✱ ✐ss✳ ✹✱ ♣♣✳ ✷✾✶✕✸✵✾✳ | ■❙❙❯❊✿ ✷ | ✷✵✶✼ | ◆♦✈❡♠❜❡r ♦♥ ❋✉t✉r✐st✐❝ ❚r❡♥❞s ♦♥ ❈♦♠♣✉t❛t✐♦♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❑♥♦✇❧❡❞❣❡ ▼❛♥❛❣❡♠❡♥t✳ ◆♦✐❞❛✿ ■❊❊❊✱ ✷✵✶✺✱ ♣♣✳ ✷✷✻✕✷✸✶✳ ❆❜♦✉t ❆✉t❤♦rs r❡❝❡✐✈❡❞ t❤❡ ▼✳❙❝✳ ❞❡❣r❡❡ ❢r♦♠ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t② ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧✲ ♦❣②✳ ❍❡ ✐s ❛ P❤✳❉✳ st✉❞❡♥t ❛t t❤❡ ❋❛❝✉❧t② ♦❢ ❊❧❡❝tr✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❬✶✷❪ ❆●❘❆❲❆▲✱ ❙✳✱ ❱✳ ❑❯▼❆❘✱ ❱❙❇✕❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❖str❛✈❛✳ ❍✐s ❑✳ P✳ ❙✳ ❘❆◆❆ ❛♥❞ P✳ ▼■❙❍❘❆✳ ❖♣t✐✲ r❡s❡❛r❝❤ ✐♥t❡r❡sts ✐♥❝❧✉❞❡ ✐♥❞✉str✐❛❧ r♦❜♦t✱ ✸❉ ♠✐③❛t✐♦♥ ♦❢ P■❉ ❝♦♥tr♦❧❧❡r ✇✐t❤ ✜rst ♦r❞❡r s✐♠✉❧❛t✐♦♥✱ ❛♥❞ ✐♥t❡❧❧✐❣❡♥t ❝♦♥tr♦❧✳ ♥♦✐s❡ ✜❧t❡r✳ ■♥✿ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ◗✉♦❝ ❇❛♦ ❉■❊P "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0)." ❝ ✷✵✶✼ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ✶✵✺ ... ✭✶✮ • sar ✿ sin ar✱ car ✿ cos ar✳ ❇❛s❡❞ ♦♥ ❊q✳ ✭✶✮ ❛♥❞ ❊q✳ ✭✷✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♣✐♣❡ R2 ❛❢t❡r t✉r♥✐♥❣ ar◦ ❛s ❊q✳ ✭✸✮✿ x22 + (y2 car − z2 sar )2 = R22 , − 21 L2 ≤ y2 sar + z2 car ≤ 12... θ3 = arctan2(s3 , c3 ) ✭✹✼✮ θi+1 ✐♥ ❞✉r❛t✐♦♥s tk ❛♥❞ tk+1 r❡s♣❡❝t✐✈❡❧②✱ t❤❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠ ♦❢ nth ❥♦✐♥t✿ ❙❡t✿ θi (t) = ai3 t3 + ai2 t2 + ai1 t + ai0 k1 = a3 c3 + a2 , k2 = a3 s3... s234 + a4 s1 c234 + a3 s1 c23 + a2 s1 c2 , ✭✸✺✮ pz = −d5 c234 + a4 s234 + a3 s23 + a2 s2 + d1 , ✭✸✻✮ θ234 = arctan2(c1 ax + s1 ay , −az ) ✭✹✷✮ ❆❢t❡r ✇❡ ❤❛✈❡ θ1 ❛♥❞ θ5 ✱ ✇❡ ❝❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡ ♦❢