Daidaita Girgiza na Delta 3D Printer tare da Dynamics Mai Canzawa Matsayi ta Amfani da B-Splines da aka Tace

1. Gabatarwa

Robobin Delta suna samun fifiko a cikin buga 3D na Fused Filament Fabrication (FFF) saboda saurinsu da suka fi na tsarin serial-axis na gargajiya. Duk da haka, wannan fa'idar saurin sau da yawa tana lalacewa saboda girgizar da ba a so wacce ke lalata ingancin sashi, matsala da ke ƙara tsananta saboda haɗakar da robot, dynamics masu dogaro da matsayi (marasa layi). Yayin da dabarun sarrafa gaba kamar Filtered B-Splines (FBS) suka yi nasarar murkushe girgiza a cikin na'urorin buga serial, amfani da su kai tsaye ga na'urorin buga delta yana da wahala a lissafa. Wannan takarda tana magance wannan matsalar ta hanyar gabatar da ingantacciyar hanyar aiwatar da ramin girgiza na tushen FBS akan na'urorin buga delta 3D.

2. Hanyoyin Aiki

Hanyar da aka gabatar tana magance ƙalubalen lissafi ta hanyar dabarar aiki guda uku da aka tsara don sanya sarrafa gaba na tushen samfurin a cikin lokaci na gaskiya ya yiwu akan masu sarrafa na'urorin buga da ke da ƙarancin albarkatu.

2.1 Tsarin Siffofin Dynamics Masu Dogaro da Matsayi a Kashe Layi

Abubuwan da ke canzawa na matsayi na samfurin motsi na robot na delta ana ƙididdige su da farko kuma a tsara su a kashe layi. Wannan ya haɗa da ƙirƙirar wakilci mai ƙarfi (misali, ta amfani da tsarin polynomial ko spline) na yadda sharuɗɗan inertia da Coriolis/centrifugal ke canzawa a cikin wurin aiki. Yayin aikin kan layi, ana iya sake gina cikakken samfurin motsi a kowane wuri cikin sauri ta hanyar kimanta waɗannan ayyuka na siffofi da aka ƙayyade da farko, maimakon ƙididdigar hadaddun kinematics da dynamics daga farko.

2.2 Lissafin Samfurin Lokaci na Gaskiya a Wuraren da aka Zana

Maimakon ƙirƙirar sabon samfurin motsi don kowane saitin maƙasudi tare da hanyar kayan aiki—tsari wanda zai yi jinkiri sosai—mai sarrafawa yana ƙididdige samfuran kawai a wuraren da aka zana da dabarun aiki tare da yanayin motsi. Ana ƙirƙirar shigarwar sarrafawa tsakanin waɗannan wuraren da aka zana ta amfani da dabarun shiga tsakani. Wannan yana rage yawan mafi girman ayyukan lissafi sosai.

2.3 Rarraba QR don Ingantacciyar Lissafi

Jigon hanyar FBS ya ƙunshi warware tsarin lissafi na layi don ƙididdige hanyar da aka tace da farko. Wannan yana buƙatar jujjuyawar matrix, wanda ke da nauyin lissafi. Takardar ta ba da shawarar amfani da rarraba QR don warware tsarin cikin inganci. Rarraba QR ($\mathbf{A} = \mathbf{Q}\mathbf{R}$) tana canza matsalar zuwa warware $\mathbf{Rx} = \mathbf{Q}^T\mathbf{b}$, wanda ya fi arha a lissafa kuma ya fi kwanciyar hankali fiye da jujjuyawar kai tsaye, musamman ga matrices masu tsari da aka saba da su a cikin wannan aikace-aikacen.

3. Cikakkun Bayanai na Fasaha & Tsarin Lissafi

Ana iya wakiltar motsin robot na delta azaman tsarin Linear Parameter-Varying (LPV) saboda inertia da haɗakar da suka dogara da matsayi. Hanyar FBS ta al'ada tana jujjuya samfurin motsi don daidaita umarnin tunani da farko. Don tsarin lokaci mai rarrabuwa, fitarwa $y[k]$ tana da alaƙa da shigarwa $u[k]$ ta hanyar aikin canja wuri. Hanyar FBS tana ƙirƙirar tacewa $F(z)$ ta yadda idan aka yi amfani da ita akan ma'anar B-spline $r[k]$, ainihin fitarwa tana bin hanyar da ake so $y_d[k]$ sosai: $y[k] \approx G(z)F(z)r[k] = y_d[k]$. Wannan yana buƙatar warware ma'auni na tacewa, wanda ya haɗa da jujjuya matrix da aka samo daga sigogin Markov na tsarin.

Kalubalen lissafi ya taso saboda ga robot na delta, samfurin shuka $G(z, \theta)$ yana bambanta da matsayi $\theta$. Matrix da za a jujjuya, $\mathbf{H}(\theta)$, ya zama mai dogaro da matsayi: $\mathbf{H}(\theta)\mathbf{f} = \mathbf{y}_d$. Hanyar da aka gabatar tana kusantar wannan a matsayin $\mathbf{H}(\theta_i)\mathbf{f} \approx \mathbf{y}_d$ a wuraren da aka zana $\theta_i$, kuma tana amfani da rarraba QR ($\mathbf{H}(\theta_i) = \mathbf{Q}_i\mathbf{R}_i$) don warware $\mathbf{f}_i$ cikin inganci a kowane samfur. Ana shiga tsakani tacewa don maki tsaka-tsaki daga waɗannan mafita da aka zana.

4. Sakamakon Gwaji & Ayyuka

4.1 Sakamakon Kwaikwayo: Saurin Lissafi

Kwaikwayoyi sun kwatanta hanyar da aka gabatar da mai sarrafa da ke amfani da ainihin samfurin LPV da aka sabunta akai-akai. Hanyar da aka gabatar—haɗa tsarin siffofi a kashe layi, zanen samfur, da rarraba QR—ta sami raguwar lokacin lissafi har sau 23, yayin da ake kiyaye daidaiton bin diddigin cikin kashi 5% na ainihin hanyar. Wannan yana nuna ingancin hanyar wajen shawo kan babban matsalar lissafi.

4.2 Tabbatar da Gwaji: Ingantaccen Bugawa & Rage Girgiza

An gudanar da gwaje-gwaje akan na'urar buga delta 3D. An kwatanta mai sarrafa da aka gabatar da mai sarrafa na asali ta amfani da samfurin Linear Time-Invariant (LTI) guda ɗaya da aka gano a wuri ɗaya a cikin wurin aiki.

• Ingancin Bugawa: Sassa da aka buga a wurare daban-daban akan farantin ginin sun nuna ingantacciyar inganci tare da mai sarrafa da aka gabatar. Siffofi sun kasance masu kaifi, tare da rage ringing da kayan tarihi na fatalwa da aka saba da su a cikin buga delta mai sauri.
• Auna Girgiza: Bayanan accelerometer da aka yi rikodin yayin bugawa sun tabbatar da tushen ingantaccen inganci. Mai sarrafa da aka gabatar ya rage girman girgiza da fiye da kashi 20% a duk faɗin wurin aiki idan aka kwatanta da mai sarrafa LTI na asali.

Bayanin Chati (An fahimta): Chati na sanduna zai iya nuna girman girgiza (a cikin g's) akan Y-axis don wuraren bugawa daban-daban (X-axis), tare da sanduna biyu a kowane matsayi: ɗaya don Mai Sarrafa LTI na Asali (mafi girma) da ɗaya don Mai Sarrafa FBS da aka Gabatar (ƙasa sosai). Wani jadawali na layi zai iya nuna lokacin lissafi a kowane yanki na yanayin motsi, yana nuna layi mai laushi, ƙasa don hanyar da aka gabatar sabanin babban layi, mai canzawa don ainihin hanyar LPV.

5. Tsarin Bincike & Misalin Lamari

Tsarin don Kimanta Yiwuwar Sarrafa Lokaci na Gaskiya:
Lokacin daidaita algorithm mai nauyin lissafi (kamar cikakken LPV FBS) don dandamali mai ƙarancin albarkatu (kamar microcontroller na tushen ARM na na'urar buga 3D), ana buƙatar bincike na tsari:

1. Gano Matsalar Tsoro: Yi bayanin algorithm don nemo ayyukan da suka fi ɗaukar lokaci (misali, jujjuyawar matrix, cikakken lissafin samfurin motsi).
2. Dabarar Kusan: Ƙayyade waɗanne lissafi za a iya kusantawa (misali, zanen samfurin sabanin sabuntawa akai-akai) ko ƙididdige su da farko (tsarin siffofi a kashe layi) tare da ƙaramin asarar aiki.
3. Ingantaccen Lissafi: Maye gurbin ayyuka na gaba ɗaya da waɗanda aka inganta don tsarin matsalar musamman (misali, rarraba QR don matrices masu tsari).
4. Tabbatarwa: Gwada sauƙaƙan algorithm da na asali a cikin kwaikwayo don amincin, sannan akan kayan aiki don aikin lokaci na gaskiya da inganci na aiki.

Misalin Lamari - Aiwatar da Tsarin:
Ga wannan aikin na'urar buga delta: Matsalar tsoro ita ce jujjuyawar matrix mai dogaro da matsayi akan layi. Dabarar kusan ita ce ƙididdigar samfuran kawai a wuraren da aka zana na yanayin motsi. Ingantaccen lissafi shine amfani da rarraba QR. Tabbatarwa ya nuna saurin sau 23 tare da kiyaye daidaito, yana tabbatar da yiwuwar.

6. Ayyuka na Gaba & Hanyoyin Bincike

• Ayyukan Robotic Masu Faɗi: Wannan hanyar aiki tana amfani da ita kai tsaye ga sauran robots masu layi daya (misali, dandamali na Stewart, tsarin kamar SCARA) da robots na serial tare da sassauƙan da ya dace da tsari, inda sarrafa tushen samfurin lokaci na gaskiya ke da wahala.
• Haɗawa da Hanyoyin Tushen Koyo: Samfurin da aka tsara siffofi a kashe layi za a iya haɓaka ko daidaita shi akan layi ta amfani da Regression na Tsarin Gaussian ko Cibiyoyin Jijiyoyi don yin la'akari da motsin da ba a ƙirƙira ba ko lalacewa, kamar yadda aka gani a cikin binciken sarrafa daidaitawa na ci gaba daga cibiyoyi kamar MIT's CSAIL.
• Haɗin gwiwar Cloud-Edge: Mafi girman nauyin lissafi na tsarin siffofi a kashe layi da shirye-shiryen gaba na yanayin motsi za a iya sauke su zuwa sabis na gajimare, tare da sauƙi na samfurin da aka zana da mai warware QR yana gudana akan na'urar gefe na na'urar buga.
• Daidaituwa a cikin Firmware: Za a iya haɗa ƙa'idodin cikin firmware na buɗe ido na na'urar buga 3D (misali, Klipper, Marlin) azaman fasali mai daraja don na'urorin buga delta da CoreXY masu sauri, tare da ba da damar samun ramin girgiza na ci gaba.

7. Nassoshi

1. Clavel, R. (1988). Delta, robot mai sauri tare da geometry mai layi daya. Proc. 18th International Symposium on Industrial Robots.
2. Briot, S., & Goldsztejn, A. (2018). Dynamics of Parallel Robots: Daga Rigid Bodies zuwa Abubuwa masu sassauƙa. Springer.
3. Okwudire, C. E., & Altintas, Y. (2009). Samfurin gauraye na tuƙi na ƙwallon ƙwallo tare da haɗakar axial, torsional, da motsi na gefe. Jaridar Mechanical Design.
4. Edoimioya, N., & Okwudire, C. (2021). Filtered B-Splines don Ramawar Girgiza akan Na'urorin Buga Serial 3D: Bita da Jagoran Aiwa. Mechatronics.
5. Codourey, A. (1998). Samfurin motsi na robots masu layi daya don aiwatar da sarrafa ƙididdiga. Jaridar Binciken Robotic ta Duniya.
6. Angel, L., & Viola, J. (2018). Fractional order PID don sarrafa ƙarfi a cikin robots na delta. Jaridar Sarrafa Injiniya da Aiwatar da Bayanai.
7. MIT Computer Science & Artificial Intelligence Laboratory (CSAIL). (2023). Tsarin Sarrafa daidaitawa da Tushen Koyo. [Kan layi]. Ana samu: https://www.csail.mit.edu

8. Bincike na Asali & Sharhin Kwararru