JP3928041B2 - Parallel mechanism for device operation and design method thereof - Google Patents

Description

の式により求める機器操作用パラレル機構の設計方法において、２つのベアリングで支持される回転連結部材における回転軸方向であるｘ軸方向の弾性係数をｋａ、回転軸方向と直角方向であるｙ，ｚ軸方向の弾性係数をｋｒ、ｘ軸周りの弾性係数を１／Φ、ｙ，ｚ軸周りの弾性係数をｋｍ、前記回転連結部材を支持するベアリングの間隔をａとするとき、前記パラレル機構について、前記各回転連結部材ｉのコンプライアンス行列Ｃｅｉを、
【数５】

の式、及び前記シリアル機構は複数の弾性体から構成され、全弾性体の弾性変位ベクトルをｅ、前記回転連結部材を含む各弾性体が接続されている各関節角度ベクトルをθとし、ｅに関するヤコビ行列をＪｅ（θ，ｅ）とし、全ての弾性体を含む前記シリアル機構の手先コンプライアンス行列Ｃｓを、
Ｃｓ＝Ｊｅ（θ，０）ＣｅＪｅ ^Ｔ （θ，０）
の式より求め、前記ｔ本のシリアル機構におけるｉ本目のシリアル機構の手先コンプライアンス行列Ｃｓｉを、
Ｃｓｉ＝Ｊｅｉ（θｉ，０）ＣｅｉＪｅｉ ^Ｔ （θ，０）
の式より求め、前記パラレル機構の手先コンプライアンスＣｐの値を所望の値となるように各係数を決定することを特徴とする。
【０００９】
また、請求項２に係る発明は、複数の部材を互いに回転自在に連結した機構の一端部を基部に回転自在に固定し、他端部を操作部に回転自在に固定したシリアル機構を複数備え、前記各シリアル機構の前記一端部を同一の基部に、前記他端部を同一の操作部に固定し、前記各シリアル機構を独立して駆動することにより前記操作部を操作する機器操作用パラレル機構の設計に際し、前記パラレル機構を構成するｔ本のシリアル機構の手先コンプライアンス行列をそれぞれＣｓｉ（ｉ＝１，・・・，ｔ）とするとき、前記パラレル機構について、手先コンプライアンス行列Ｃｐを、
【数４】

の式により求める機器操作用パラレル機構の設計方法において、回転連結部材である前記第１ベアリング、第２ベアリング、第３ベアリング、第４ベアリングにおけるコンプライアンス行列を各々Ｃ e １、Ｃ e ２、Ｃ e ３、Ｃ e ４とし、２つのベアリングで支持される回転連結部材における回転軸方向であるｘ軸方向の弾性係数をｋａ、回転軸方向と直角方向であるｙ，ｚ軸方向の弾性係数をｋｒ、ｘ軸周りの弾性係数を１／Φ、ｙ，ｚ軸周りの弾性係数をｋｍ、前記回転連結部材を支持するベアリングの間隔をａとするとき、各ベアリングのコンプライアンス行列を、
【数６】

の式、及び前記シリアル機構は複数の弾性体から構成され、全弾性体の弾性変位ベクトルをｅ、前記回転連結部材を含む各弾性体が接続されている各関節角度ベクトルをθとし、ｅに関するヤコビ行列をＪｅ（θ，ｅ）とし、全ての弾性体を含む前記シリアル機構の手先コンプライアンス行列Ｃｓを、
Ｃｓ＝Ｊｅ（θ，０）ＣｅＪｅ ^Ｔ （θ，０）
の式より求め、前記ｔ本のシリアル機構におけるｉ本目のシリアル機構の手先コンプライアンス行列Ｃｓｉを、
Ｃｓｉ＝Ｊｅｉ（θｉ，０）ＣｅｉＪｅｉ ^Ｔ （θ，０）
の式より求め、
該式に実際のベアリングの値を入れることにより各ベアリングのコンプライアンス行列の値を求め、
前記パラレル機構の手先コンプライアンスＣｐの値を所望の値となるように各係数を決定し、且つ、全てのベアリングのΦを、
１×１０^−２＜Φ＜１×１０^６ （ｒａｄ／Ｎｍｍ）
の領域に選定することを特徴とする。
【００１５】
【発明の実施の形態】
本発明は上記のような従来技術の課題を解決するため、最初、パラレル機構の基本構成をなす図２に示すモデルについて検討を行う。このモデルについては、基部としてのベース４２と操作部としての移動板４４の間に、複数のリンクとベアリングで構成されるシリアルマニピュレータ４３を複数本パラレルに設けることにより、移動板４４をベースに対して移動可能に支持した構成をなし、図中においてはこのシリアルマニピュレータ４３を、１、２・・・、ｔのｔ本用いた例を示している。
【００１６】
このモデルを解析するに際して、最初、ここで用いられるシリアルマニピュレータ４３の単体について解析を行い、その結果に基づいてこれを統合し、上記パラレルモデルの剛性について検討を行う。一般にシリアルマニピュレータ４３は図１においてシリアルマニピュレータ４０として示すように、ｍ個の弾性変形する要素Ｌ_１，Ｌ_２，・・・Ｌ_ｍと、ｎ個の関節Ｂ１，Ｂ２・・・Ｂｎから構成され、その基部をベースに回転自在に固定し、先端に操作部４１を設けている。なお、以下の解析に際して、個々に弾性変形する要素を各弾性体、弾性変形する全ての要素をまとめて全弾性体と呼ぶ。
【００１７】
各弾性体は、力、モーメントが作用することにより位置、姿勢が弾性変形するため、各弾性体ｉの弾性変位ベクトルをそれぞれｅ_ｉ（ｉ＝１，・・・，ｍ）とすると、
ｅ_ｉ＝［δ_ｘｉ δ_ｙｉ δ_ｚｉ Φ_ｘｉ Φ_ｙｉ Φ_ｚｉ］^Ｔ （１）
となる。ここで、δ_ｘｉ、δ_ｙｉ、δ_ｚｉはそれぞれ各弾性体ｉの各軸方向の弾性変位である。また、Φ_ｘｉ、Φ_ｙｉ、Φ_ｚｉはそれぞれ各弾性体ｉの各軸周りの弾性変位である。したがって、全弾性体の弾性変位ベクトルｅは、
ｅ＝［ｅ_１ ^Ｔ ｅ_２ ^Ｔ ・・・ ｅ_ｍ ^Ｔ］^Ｔ （２）
と表すことができる。
【００１８】
各弾性体ｉに力・モーメントベクトルｆ_ｌｉが加わるときの全弾性体の弾性変位ベクトルは次のように表される。
ｅ＝Ｃ_ｅ［ｆ_ｌ１ ^Ｔ ｆ_ｌ２ ^Ｔ ・・・ ｆ_ｌｍ ^Ｔ］^Ｔ （３）
Ｃ_ｅ＝diag［C_ｅ１ Ｃ_ｅ２ ・・・ Ｃ_ｅｍ］ （４）
ここで、Ｃ_ｅは各弾性体の構造的性質によって決定される全弾性体のコンプライアンス行列であり、Ｃ_ｅｉは各弾性体のローカルなコンプライアンス行列である。
【００１９】
各弾性体が接続されている各関節角度ベクトルをθとして、ｅに関するヤコビ行列をＪ_ｅ（θ，ｅ）とすると、全ての弾性体を含むシリアルマニピュレータのコンプライアンス行列Ｃ_ｓは、
Ｃ_ｓ＝Ｊ_ｅ（θ,０）Ｃ_ｅＪ_ｅ ^Ｔ（θ,０） （５）
と導出される。また、コンプライアンス行列Ｃ_ｓの左上３×３小行列から、手先の力に対する位置のコンプライアンス楕円体、即ち単位力ベクトルが作用した際の手先弾性変位ベクトルの軌跡が得られる。
【００２０】
上記のように導出したシリアルマニピュレータのコンプライアンス行列を用いて、パラレル機構のコンプライアンス行列を導出する。
【００２１】
図２にパラレル機構のモデルを示す。このパラレル機構は図１に示すシリアルマニピュレータｔ本から構成される。したがって、ｔ本のシリアルマニピュレータの手先コンプライアンス行列をそれぞれＣ_ｓｉ（ｉ＝１・・・，ｔ）とすると、パラレル機構の手先コンプライアンス行列Ｃ_ｐは、
【数１３】

と導出される。ここで、ベース、移動板は弾性変形しないと仮定している。この式により前記のような構造のパラレル機構における剛性を求めることができ、この剛性を考慮してパラレル機構の適切な設計を行うことができる。また、適切な剛性を有するパラレル機構とすることができる。
【００２２】
次に、リンクのモデル化、特に、リンクのコンプライアンス行列について検討する。図３に示すリンクｉの先端に力Ｆ_ｘｉ、Ｆ_ｙｉ、Ｆ_ｚｉ、モーメントＭ_ｘｉ、Ｍ_ｙｉ、Ｍ_ｚｉが作用すると、手先の変位は、
【数１４】

となる。ここで、Ｌはリンクの長さ、Ｅは縦弾性係数、Ｇは横弾性係数である。また、Ｉ_ｘ、Ｉ_ｙ、Ｉ_ｚはそれぞれｘ，ｙ，ｚ軸に対する断面２次モーメント、Ｉ_ｐは極断面２次モーメントである。したがって、各弾性体（リンク）ｉのコンプライアンス行列Ｃ_ｅｉは、
【数１５】

となる。
【００２３】
次にベアリングのモデル化、特に、ベアリングのコンプライアンス行列について検討する。
ベアリングｉのコンプライアンス行列Ｃ_ｅｉは、
【数１６】

となる。ここで、ｋ_ａはｘ軸方向の弾性係数、ｋ_ｒはｙ、ｚ軸方向の弾性係数、１／Φはｘ軸周りの弾性係数、ｋ_ｍはｙ，ｚ軸周りの弾性係数である。また、回転軸はｘ軸と平行としている。
【００２４】
受動軸にベアリングが使用される場合、ベアリングの回転軸は受動軸と一致し、回転軸の弾性係数は零、Φは無限大の値となる。しかし、数値計算を行うためにはΦをある定数と仮定しなければならない。そこで、ここではΦをコンプライアンス行列の他の要素より十分に大きい値と仮定し、数値計算で使用している。実際に採用する値については、後に詳述する。
【００２５】
次に改良型デルタ機構のコンプライアンス行列について検討する。ここでは、先に述べたパラレル機構の剛性解析手法を、以前より本発明者等が研究開発してきた小型６自由度ハプテックインタフェースの並進部である改良型デルタ機構に適用した例を示す。この改良型デルタ機構については、図４に示すようなシリアル機構を３個用い、図５に示すようなパラレル機構を構成したものである。
【００２６】
図４において、基部としての基板１０に固定される基板側取付部材１１は１対のベアリング１２，１２により第１回転軸１３を支持しており、この第１回転軸１３はその端部に設けたモータ１４によって回転駆動される。第１回転軸１３にはアーム１５の図中下端部が固定され、その上端部には第１ベアリングとしての１対のベアリング１６，１６により、リンク機構の第１の構成部材としての第２回転軸１７が回転自在に支持されている。第２回転軸１７の両端部にはこの第２回転軸１７の軸線と直角に第１連結支軸１８と第２連結支軸１９とを突出固定し、第１連結支軸１８にはリンク機構を構成する第２の部材としての第１リンクロッド２０の図中下端部を、また第２連結支軸１９にはリンク機構を構成する第３の部材としての第２リンクロッド２１の下端部を、各々第２ベアリングとしての１対のベアリング２２，２２、２３，２３により回転自在に支持している。
【００２７】
第１リンクロッド２０の図中上端部は第３連結支軸２４に、また第２リンクロッド２１の上端部は第４連結支軸２５に、各々第３ベアリングとしての１対のベアリング２６，２６、２７，２７により回転自在に支持されている。更に、前記第３連結支軸２４及び第４連結支軸２５はリンク機構を構成する第４の部材としての第３回転軸２８に対して、その軸線と直角に固定され、前記第３回転軸２８は移動板３０に固定されている移動板側固定部材２９に対して、第４ベアリングとしての１対のベアリング３１，３１により回転自在に支持されている。
【００２８】
上記のような第１リンクロッド２０、第２リンクロッド２１、第１連結支軸１８と第２連結支軸１９を固定した第２回転軸１７、第３連結支軸２４と第４連結支軸２５を固定した第３回転軸２８によって平行リンク機構３２を構成している。また、基板１０に固定される基板側取り付け部材１１に回転自在に支持されたアーム１５と、操作部としての移動板３０に固定された前記平行リンク機構３２を互いに揺動自在に結合することによりシリアル機構３３を構成している。
【００２９】
上記構成により、モータ１４を回転するとアーム１５が回転し、第１連結支軸１８と第２連結支軸１９が固定された第２回転軸１７が第１回転軸１３と平行に上下動する。それにより、もしも移動板３０が全く自由に移動するときには平行リンク機構３２は一体的に上下動し、移動板を上下動することとなるが、移動板３０が何らかの拘束を受けているときには、その拘束の状態に応じて平行リンク機構３２の第１リンクロッド２０と第２リンクロッド２１が平行に揺動し、平行リンク機構３２の変形作動が行われる。
【００３０】
図５に示す改良型デルタ機構は上記のような構造のシリアル機構をなすシリアル機構３３を３個並列に設けたものであり、図４においては同一の基板１０に固定された第１シリアル機構３４、第２シリアル機構３５、第３シリアル機構３６の図中上端部は、各々同一の移動板３０に固定され、それにより各シリアル機構のモータを外部の制御装置により独立して駆動することによって、移動板２０に固定された作動部３７を任意の態様で移動することができるようにしている。
【００３１】
したがって、この装置をパラレルマニピュレータとして用いるときには、この装置から離れた場所にあるマニュアル操作部材を利用者が操作することにより、制御装置がその操作信号によって前記各モータを駆動し、操作部材の操作に対応して作動部３７を任意の位置に移動させることができるようになる。
【００３２】
上記のように、本発明者等が研究開発を行っている改良型デルタ機構においては、２本のロッド２０，２１、１対の第２ベアリング２２，２３、１対の第３ベアリング２６，２７から構成されるロッド部は平面パラレル機構となっており、また、全ての回転軸にはベアリングがペアで設置されている。
【００３３】
上記のような機構の剛性の解析に際して、この機構において１対で使用されるベアリングのモデル化を検討する。図６に示すような平行に配置された二つのベアリング（Bearing A, Bearing B)の中間点Ｏでのコンプライアンス行列を考える。ベアリング間距離をａとする。ｘ、ｙ、ｚ軸方向の弾性係数は、前記図１及び図２のモデルにおいて採用している一つのベアリングに対するコンプライアンス行列の２倍となり、ｘ軸周りの弾性係数も２倍となる。しかし、ｙ、ｚ軸周りの弾性係数は少し複雑となる。図７には点Ｏにモーメントが作用した場合のモデルを示す。Ｍとθの関係は、
【数１７】

となる。以上より、点Ｏにおいて二つのベアリングから構成される各弾性体（ベアリング）ｉのコンプライアンス行列は、
【数１８】

となる。但し回転軸はｘ軸方向と一致するものとする。したがって、特にパラレル機構で用いられる各シリアル機構において１対のベアリングが使用されるとき、この式に基づいてベアリング部分の剛性を求めることができ、この剛性を考慮してパラレル機構の適切な設計を行うことができる。また、適切な剛性を有するパラレル機構とすることができる。
【００３４】
次に、上記のようにして求めた本発明による剛性解析手法が、パラレル機構の設計指標として使用できるかを評価するための実験を行ったのでその結果について述べる。本実験で使用する改良型デルタ機構は前記図４、図５に示すものと同様の機構を用いた。
【００３５】
本実験における座標系の原点は図５における点Ｏとする。まず、点Ａにｘ軸に平行な力（9.8［Ｎ］）を付加し、点Ｂにおける変位を測定し、次いで、実機の測定結果と剛性解析手法に基づく計算結果を比較する。また、本実験は表１に示すような２種類のパラメータ（Type I, Type II)で行った。なお、本実験では、アームを固定して行うため、図４の第１ベアリング１２，１２に相当する部分及びアームによる弾性変形は考慮しない。
【表１】

【００３６】
二つのパラメータ（Type I, Type II)での実機による結果と数値解析に基づく計算結果を図８、図９に示す。同図から明らかなように、実機による結果と計算結果の傾向はほぼ一致している。誤差の原因は、実機に存在するガタの影響、実機を構成している各要素の弾性係数は荷重により変化するが数値解析では一定と仮定したことなどが考えられる。以上の結果から、本発明による剛性解析手法は、パラレル機構の設計指標としては十分に使用可能であるといえる。
【００３７】
次に、上記のようなベアリングの回転軸に関する弾性係数のモデル化について検討する。ベアリングが受動軸に使用される場合、Φは無限大となる。しかし、数値計算を行うためには、Φを無限大として取り扱うことはできない。そこで、ここでは、Φをコンプライアンス行列の他の要素より十分に大きい値と仮定する。以下このΦについて考察する。
【００３８】
ハプティクインタフェースの前記改良型デルタ機構で採用したベアリング１、２、３、４におけるコンプライアンス行列の各要素は、
【数１９】

【数２０】

である。ここで、並進に対する要素の単位は［ｍｍ／Ｎ］、回転に対する要素の単位は［rad／Ｎmm］である。また、図４における第１ベアリング１２，１２はモータの駆動軸と連結されているため、Φの値はゼロと仮定する。
【００３９】
図１０にΦの値を変化させることにより、前記Type IIでの手先コンプライアンス行列を構成する各要素の変化を示す。図中の記号は、
【数２１】

で示される各要素に対応する。
【００４０】
全ての各要素は、Φ＜１×１０^−７と１×１０^−２＜Φ＜１×１０^６の領域で安定した値となっている。Φ＜１×１０^−７は回転軸が固定された状態１×１０^−２＜Φ＜１×１０^６は回転軸が固定されていない状態と等価であると考えられる。ここで、Φの値を１×１０^５とすると、この値は、他の要素より十分に大きく受動軸の弾性係数を表しているといえる。
【００４１】
上記のように、パラレル機構は、一般に高精度、高速、高剛性の特徴を有している。しかし、各部材やベアリングなどの弾性変形による微少な変形が、手先の剛性に大きな影響を与える。また、手先の剛性は各部材及びベアリングのサイズや配置場所によっても大きく影響を受ける。
【００４２】
そのため、本発明はパラレル機構を構成するベアリングを含む各部材の弾性変形を個別にモデル化し、これらのモデルに基づいた構造剛性解析手法を提案したものである。この剛性解析の特徴は、ベアリングを含む機構の構成要素を個別にモデル化した後、構成要素を統合してパラレル機構のモデル化を行う手法であるため、個々の機構に依存したものとなっていない。そのため、シリアルマニピュレータ、パラレルマニピュレータを問わず、弾性変形する多くの機構に適用することができ、汎用性に優れたものとすることができ、特に剛性に関する解析手法が定まっていないパラレルマニピュレータにとってきわめて有効である。
【００４３】
更に本発明をパラレル機構の一つである改良型デルタ機構を、前記小型６自由度ハプティックインタフェースの並進部として用いた実機に対して、本発明による剛性解析に基づく計算を行った結果と実際との比較評価実験を行った結果、本解析手法が十分に設計指標として使用できることが検証された。
【００４４】
【発明の効果】
本発明は上記のように構成したので、各種機器を操作する装置に用いるパラレル機構において、正確な解析データ手法を得ることにより剛性の高い機器操作用パラレル機構の設計方法を提供することができる。また、その解析データによりハプティックインタフェースとして実質的に高剛性の機器操作用パラレル機構を得ることができる。特に、パラレル機構を構成するベアリングを含む各部材の弾性変形を個別にモデル化し、これらのモデルに基づいた構造剛性解析手法を得ることにより剛性の高いベアリングを含む機器操作用のパラレル機構の汎用性の高い設計方法を提供し、また、その解析データにより実質的に高剛性の機器操作用のベアリングを含むパラレル機構を得ることができる。
【図面の簡単な説明】
【図１】 シリアル機構の模式図である。
【図２】パラレル機構の模式図である。
【図３】 リンクの先端に力とモーメントが作用したときの説明図である。
【図４】本発明によるパラレルインターフェースにおける改良型デルタ機構に用いるシリアル機構の斜視図である。
【図５】本発明によるパラレルインターフェースにおける改良型デルタ機構の斜視図である。
【図６】本発明におけるパラレル機構に用いられる１対のベアリングの剛性解析用平面図である。
【図７】同、１対のベアリングの剛性解析用模式図である。
【図８】本発明を改良型デルタ機構に適用することにより得られた値と、実際の値との比較実験において、TypeIの条件における結果を示すグラフである。
【図９】本発明を改良型デルタ機構に適用することにより得られた値と、実際の値との比較実験において、TypeIIの条件における結果を示すグラフである。
【図１０】回転方向の弾性係数を変化させたときの手先コンプライアンス行列を構成する各要素の変化を示すグラフである。
【符号の説明】
１０ 基板
１１ 基板側取付部材
１２ ベアリング
１３ 第１回転軸
１４ モータ
１５ アーム
１６ ベアリング
１７ 第２回転軸
１８ 第１連結支軸
１９ 第２連結支軸
２０ 第１リンクロッド
２１ 第２リンクロッド
２２、２３ ベアリング
２４ 第３連結支軸
２５ 第４連結支軸
２６、２７ ベアリング
２８ 第３回転軸
２９ 移動板側固定部材
３０ 移動板
３１ ベアリング
３２ 平行リンク機構
３３ アーム・ロッド結合体[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a parallel mechanism used for a haptic interface for operating various devices, and in particular, based on the result of accurate analysis of the rigidity of a parallel mechanism having a bearing, the rigidity of the operation mechanism using the parallel mechanism can be increased. The present invention relates to a highly rigid parallel mechanism and a design method thereof.
[0002]
[Prior art]
For example, in order to assemble a large structure such as a space station, work such as rendezvous, docking, or object capture is necessary, and the work can be performed by accurately operating various devices remotely. There is a need to do, and research and development is underway. Become more. In various other medical devices and experimental devices, these various devices are often remotely controlled, and manipulators have been conventionally used for the operations, and in particular, haptics that can be appropriately operated by the operator's sense of touch. Research and development of the interface is underway.
[0003]
For example, a plurality of links L ₁ , L ₂ ... L _m as shown in FIG. 1 are connected to joints B ₁ , B _2. A serial manipulator 40 that is connected in series with _Bn to form a manipulator may be used. However, since this method has problems such as a decrease in accuracy due to the accumulation of position errors and an increase in weight for maintaining the strength by the cantilever support mechanism, as a more rigid mechanism, for example, as shown in FIG. , A parallel mechanism manipulator in which a plurality of serial manipulators 43, 43,... Are extended and a movable plate 44 as an object to be operated is supported at their tips may be used. This manipulator is capable of high speed, high accuracy, high load, high rigidity and light weight, and is expected to be used in various fields by taking advantage of its features.
[0004]
Such a parallel mechanism manipulator can form a relatively high-rigidity mechanism with a light weight. However, even when such a parallel mechanism is used, a more direct tactile sensation is presented to the operator. Therefore, a parallel mechanism with higher accuracy, higher speed, and higher rigidity is required. In particular, in order to increase the degree of freedom of the manipulator, it is often necessary to connect a plurality of arms and rods, but it is necessary to provide a bearing at the joint as the connection, and the elastic deformation of this bearing is also remotely controlled. This greatly affects the rigidity of the entire manipulator.
[0005]
On the other hand, since the parallel mechanism is required to be further reduced in size and weight, it is difficult to increase the rigidity in terms of structure. As a countermeasure, the characteristics related to the rigidity of the parallel mechanism to be used in advance are obtained accurately, and the value that takes the characteristics into account for the amount of operation by the operator is corrected and output. The mechanism can be substantially the same as the parallel mechanism.
[0006]
[Problems to be solved by the invention]
For this reason, research is being conducted to model structural analysis by modeling elastic deformation related to the parallel mechanism as described above. However, the structural analysis is not yet sufficient, and a method for increasing the rigidity of the manipulator using an actual parallel mechanism based on the analysis data has not yet been developed. In particular, many parallel mechanisms use bearings in the parts corresponding to the joints as described above, and it is necessary to model them directly as a whole, including the elastic deformation of these bearings, and perform rigidity analysis. On the other hand, there has not yet been proposed a method for raising such rigidity analysis to a practical level.Therefore, there is a demand for an appropriate rigidity analysis method that takes into account the elastic deformation of individual bearings. Development of a parallel mechanism for device operation using such a stiffness analysis method has been desired.
[0007]
Therefore, the present invention provides a method for designing a rigid parallel mechanism for device operation by obtaining an accurate analysis data method in a parallel mechanism used in an apparatus for operating various devices, and substantially uses the analysis data to The main purpose is to provide a parallel mechanism for high-rigidity equipment operation. In addition, the elastic deformation of each member including the bearings constituting the parallel mechanism is individually modeled, and a structural rigidity analysis method based on these models is obtained. The main purpose is to provide a parallel mechanism design method for equipment operation including bearings with high rigidity, and to obtain a parallel mechanism including bearings for equipment operation with high rigidity as a haptic interface based on the analysis data. And
[0008]
[Means for Solving the Problems]
In order to solve the above-mentioned problem, in the invention according to claim 1, one end portion of a mechanism in which a plurality of members are rotatably connected to each other is fixed to the base portion, and the other end portion is rotatably fixed to the operation portion. A plurality of serial mechanisms are provided, and the one end of each serial mechanism is fixed to the same base and the other end is fixed to the same operation unit, and the operation units are operated by independently driving the serial mechanisms. When designing the parallel mechanism for device operation, the hand compliance matrix of the t serial mechanisms constituting the parallel mechanism is Csi (i = 1,..., T), respectively. The matrix Cp
[Expression 4]

In the design method of the parallel mechanism for device operation obtained by the formula : ## EQU1 ## where k is the elastic coefficient in the x-axis direction, which is the rotation axis direction, and y, z are the directions perpendicular to the rotation axis direction. When the elastic coefficient in the axial direction is kr, the elastic coefficient around the x-axis is 1 / Φ, y, the elastic coefficient around the z-axis is km, and the distance between the bearings supporting the rotary connecting member is a, the parallel mechanism , And a compliance matrix Cei for each of the rotational connecting members i,
[Equation 5]

And the serial mechanism is composed of a plurality of elastic bodies, where e is the elastic displacement vector of all elastic bodies, and θ is each joint angle vector to which each elastic body including the rotation connecting member is connected. The Jacobian matrix is Je (θ, e), and the hand compliance matrix Cs of the serial mechanism including all elastic bodies is
Cs = Je (θ, 0) CeJe ^T (θ, 0)
And the tip compliance matrix Csi of the i-th serial mechanism in the t serial mechanisms,
Csi = Jei (θi, 0) CeiJei ^T (θ, 0)
Each coefficient is determined so that the value of the tip compliance Cp of the parallel mechanism becomes a desired value.
[0009]
Further, the invention according to claim 2 includes a plurality of serial mechanisms in which one end portion of a mechanism in which a plurality of members are rotatably connected to each other is fixed to the base portion and the other end portion is rotatably fixed to the operation portion. The one end of each serial mechanism is fixed to the same base, the other end is fixed to the same operation unit, and the operation units are operated by independently driving the serial mechanisms. In designing the mechanism, when the hand compliance matrixes of the t serial mechanisms constituting the parallel mechanism are Csi (i = 1,..., T), the hand compliance matrix Cp for the parallel mechanism is
[Expression 4]

In the design method of the parallel mechanism for device operation obtained by the following equation , the compliance matrices in the first bearing, the second bearing, the third bearing, and the fourth bearing, which are rotational connection members, are represented by C e 1, C e 2, C e, respectively. 3 and C e 4, the elastic coefficient in the x-axis direction that is the rotational axis direction of the rotary coupling member supported by the two bearings is ka, and the elastic modulus in the y- and z-axis directions that are perpendicular to the rotational axis direction is kr. , Where the elastic coefficient around the x-axis is 1 / Φ, the elastic coefficient around y, the z-axis is km, and the distance between the bearings supporting the rotary connecting member is a, the compliance matrix of each bearing is
[Formula 6]

And the serial mechanism is composed of a plurality of elastic bodies, where e is the elastic displacement vector of all elastic bodies, and θ is each joint angle vector to which each elastic body including the rotation connecting member is connected. The Jacobian matrix is Je (θ, e), and the hand compliance matrix Cs of the serial mechanism including all elastic bodies is
Cs = Je (θ, 0) CeJe ^T (θ, 0)
And the tip compliance matrix Csi of the i-th serial mechanism in the t serial mechanisms,
Csi = Jei (θi, 0) CeiJei ^T (θ, 0)
From the equation
Obtain the value of the compliance matrix for each bearing by putting the actual bearing values into the equation ,
Each coefficient is determined so that the value of the tip compliance Cp of the parallel mechanism becomes a desired value, and Φ of all the bearings is
1 × 10 ⁻² <Φ <1 × 10 ⁶ (rad / Nmm)
It is characterized in that it is selected in the area.
[0015]
DETAILED DESCRIPTION OF THE INVENTION
In order to solve the above-described problems of the prior art, the present invention first examines the model shown in FIG. 2 which forms the basic configuration of the parallel mechanism. For this model, a plurality of serial manipulators 43 composed of a plurality of links and bearings are provided in parallel between a base 42 as a base and a moving plate 44 as an operation unit, so that the moving plate 44 is attached to the base. In the figure, the serial manipulator 43 is shown using t, 1, 2,..., T.
[0016]
When analyzing this model, first, the single unit of the serial manipulator 43 used here is analyzed and integrated based on the result, and the rigidity of the parallel model is examined. Generally, the serial manipulator 43 is composed of m elastically deforming elements L ₁ , L ₂ ,... L _m and n joints B 1, B 2,... Bn, as shown as a serial manipulator 40 in FIG. The base portion is rotatably fixed to the base, and the operation portion 41 is provided at the tip. In the following analysis, elements that are elastically deformed individually are referred to as each elastic body, and all elements that are elastically deformed are collectively referred to as a total elastic body.
[0017]
Since each elastic body is elastically deformed in position and posture due to the action of force and moment, if the elastic displacement vector of each elastic body i is e _i (i = 1,..., M),
e _i = [δ _xi δ _yi δ _zi Φ _xi Φ _yi Φ _zi ] ^T (1)
It becomes. Here, δ _xi , δ _yi , and δ _zi are elastic displacements of the respective elastic bodies i in the respective axial directions. Further, Φ _xi , Φ _yi , and Φ _zi are elastic displacements around the respective axes of the respective elastic bodies i. Therefore, the elastic displacement vector e of the entire elastic body is
_{^{e = [e 1 T e 2}} T ··· e m T] T (2)
It can be expressed as.
[0018]
The elastic displacement vector of all elastic bodies when a force / moment vector _fl i is applied to each elastic body i is expressed as follows.
_{^{e = C e [f l1 T}} f l2 T ··· f lm T] T (3)
C _e = diag [C _e1 C _e2 ... C _em ] (4)
Here, C _e is a compliance matrix of all elastic bodies determined by the structural properties of each elastic body, and C _ei is a local compliance matrix of each elastic body.
[0019]
Assuming that each joint angle vector to which each elastic body is connected is θ and the Jacobian matrix for _e is J _e (θ, e), the compliance matrix C _s of the serial manipulator including all elastic bodies is
C _s = J _e (θ, 0) C _e J _e ^T (θ, 0) (5)
Is derived. In addition, from the upper left 3 × 3 small matrix of the compliance matrix C _s , the compliance ellipsoid of the position with respect to the hand force, that is, the locus of the hand elastic displacement vector when the unit force vector acts is obtained.
[0020]
The compliance matrix of the parallel mechanism is derived using the compliance matrix of the serial manipulator derived as described above.
[0021]
FIG. 2 shows a model of the parallel mechanism. This parallel mechanism is composed of t serial manipulators shown in FIG. Therefore, if the hand compliance matrices of the t serial manipulators are C _si (i = 1..., T), the hand compliance matrix C _p of the parallel mechanism is
[Formula 13]

Is derived. Here, it is assumed that the base and the moving plate are not elastically deformed. With this equation, the rigidity of the parallel mechanism having the above-described structure can be obtained, and the parallel mechanism can be appropriately designed in consideration of this rigidity. Moreover, it can be set as the parallel mechanism which has appropriate rigidity.
[0022]
Next, we consider link modeling, especially the link compliance matrix. When the forces F _xi , F _yi , F _zi , moments M _xi , M _yi , M _zi act on the tip of the link i shown in FIG.
[Expression 14]

It becomes. Here, L is the length of the link, E is the longitudinal elastic modulus, and G is the transverse elastic modulus. Further, I _x , I _y , and I _z are cross-sectional secondary moments with respect to the x, y, and z axes, respectively, and I _p is a polar cross-sectional secondary moment. Therefore, the compliance matrix C _ei of each elastic body (link) i is
[Expression 15]

It becomes.
[0023]
Next, we will consider bearing modeling, especially the bearing compliance matrix.
The compliance matrix C _{ei for} bearing i is
[Expression 16]

It becomes. Here, k _a is the elastic coefficient of the x-axis direction, k _r is y, z-axis direction of the elastic modulus, 1 / [Phi elastic modulus about the x-axis, k _m is the modulus of elasticity of about y, z-axis. The rotation axis is parallel to the x axis.
[0024]
When a bearing is used for the passive shaft, the rotational axis of the bearing coincides with the passive shaft, the elastic coefficient of the rotational shaft is zero, and Φ is an infinite value. However, in order to perform numerical calculations, Φ must be assumed to be a constant. Therefore, here, Φ is assumed to be sufficiently larger than other elements of the compliance matrix, and is used in the numerical calculation. The values actually employed will be described in detail later.
[0025]
Next, the compliance matrix of the improved delta mechanism is discussed. Here, an example in which the parallel mechanism stiffness analysis method described above is applied to an improved delta mechanism, which is a translation part of a small 6-DOF haptic interface that has been researched and developed by the present inventors, has been shown. The improved delta mechanism uses three serial mechanisms as shown in FIG. 4 and constitutes a parallel mechanism as shown in FIG.
[0026]
In FIG. 4, a board-side mounting member 11 fixed to a base board 10 as a base part supports a first rotating shaft 13 by a pair of bearings 12 and 12, and this first rotating shaft 13 is provided at an end portion thereof. The motor 14 is driven to rotate. A lower end portion of the arm 15 in the figure is fixed to the first rotating shaft 13, and a second rotation as a first component member of the link mechanism is provided at the upper end portion by a pair of bearings 16, 16 as a first bearing. A shaft 17 is rotatably supported. A first connection support shaft 18 and a second connection support shaft 19 are projected and fixed at both ends of the second rotation shaft 17 at right angles to the axis of the second rotation shaft 17. The lower end portion of the first link rod 20 as a second member constituting the link mechanism in the figure, and the lower end portion of the second link rod 21 as the third member constituting the link mechanism on the second connecting support shaft 19. These are supported rotatably by a pair of bearings 22, 22, 23, 23 as second bearings.
[0027]
The upper end of the first link rod 20 in the drawing is the third connecting support shaft 24, and the upper end of the second link rod 21 is the fourth connecting support shaft 25, and a pair of bearings 26, 26 as third bearings, respectively. , 27, 27 are rotatably supported. Further, the third connecting support shaft 24 and the fourth connecting support shaft 25 are fixed to the third rotating shaft 28 as the fourth member constituting the link mechanism at right angles to the axis thereof, and the third rotating shaft. Reference numeral 28 denotes a movable plate side fixed member 29 fixed to the movable plate 30, and is rotatably supported by a pair of bearings 31, 31 as fourth bearings.
[0028]
The first link rod 20, the second link rod 21, the first connection support shaft 18 and the second connection support shaft 19 fixed to the second rotary shaft 17, the third connection support shaft 24 and the fourth connection support shaft as described above. A parallel link mechanism 32 is constituted by the third rotating shaft 28 to which 25 is fixed. In addition, the arm 15 rotatably supported by the substrate side mounting member 11 fixed to the substrate 10 and the parallel link mechanism 32 fixed to the moving plate 30 as the operation unit are coupled to each other so as to be swingable. A serial mechanism 33 is configured.
[0029]
With the above configuration, when the motor 14 is rotated, the arm 15 is rotated, and the second rotating shaft 17 to which the first connecting support shaft 18 and the second connecting support shaft 19 are fixed moves up and down in parallel with the first rotating shaft 13. Thereby, if the moving plate 30 moves completely freely, the parallel link mechanism 32 moves up and down integrally and moves the moving plate up and down. However, when the moving plate 30 is subjected to some restriction, The first link rod 20 and the second link rod 21 of the parallel link mechanism 32 swing in parallel according to the restrained state, and the parallel link mechanism 32 is deformed.
[0030]
The improved delta mechanism shown in FIG. 5 is provided with three serial mechanisms 33 forming a serial mechanism having the above structure in parallel. In FIG. 4, the first serial mechanism 34 fixed to the same substrate 10 is used. The upper ends of the second serial mechanism 35 and the third serial mechanism 36 in the figure are fixed to the same moving plate 30, respectively, thereby independently driving the motor of each serial mechanism by an external control device, The operating portion 37 fixed to the moving plate 20 can be moved in an arbitrary manner.
[0031]
Therefore, when this device is used as a parallel manipulator, a user operates a manual operation member located away from this device, so that the control device drives each of the motors by the operation signal to operate the operation member. Correspondingly, the operating part 37 can be moved to an arbitrary position.
[0032]
As described above, in the improved delta mechanism that the inventors have been researching and developing, the two rods 20, 21, a pair of second bearings 22, 23, and a pair of third bearings 26, 27. The rod part comprised from is a plane parallel mechanism, and bearings are installed in pairs on all rotating shafts.
[0033]
When analyzing the rigidity of the mechanism as described above, the modeling of a pair of bearings used in this mechanism will be considered. Consider a compliance matrix at an intermediate point O between two bearings (Bearing A, Bearing B) arranged in parallel as shown in FIG. Let the distance between bearings be a. The elastic modulus in the x, y, and z axis directions is twice the compliance matrix for one bearing employed in the models of FIGS. 1 and 2, and the elastic coefficient around the x axis is also twice. However, the elastic coefficients around the y and z axes are a little complicated. FIG. 7 shows a model when a moment acts on the point O. The relationship between M and θ is
[Expression 17]

It becomes. From the above, the compliance matrix of each elastic body (bearing) i composed of two bearings at the point O is
[Formula 18]

It becomes. However, the rotation axis coincides with the x-axis direction. Therefore, especially when a pair of bearings are used in each serial mechanism used in the parallel mechanism, the rigidity of the bearing portion can be obtained based on this formula, and an appropriate design of the parallel mechanism can be determined in consideration of this rigidity. It can be carried out. Moreover, it can be set as the parallel mechanism which has appropriate rigidity.
[0034]
Next, an experiment for evaluating whether the rigidity analysis method according to the present invention obtained as described above can be used as a design index for a parallel mechanism will be described. The improved delta mechanism used in this experiment was the same as that shown in FIGS.
[0035]
The origin of the coordinate system in this experiment is point O in FIG. First, a force parallel to the x axis (9.8 [N]) is applied to point A, the displacement at point B is measured, and then the measurement result of the actual machine is compared with the calculation result based on the stiffness analysis method. In addition, this experiment was performed with two types of parameters (Type I and Type II) as shown in Table 1. In this experiment, since the arm is fixed, elastic deformation due to the portion corresponding to the first bearings 12 and 12 in FIG. 4 and the arm is not considered.
[Table 1]

[0036]
FIG. 8 and FIG. 9 show the results of the actual machine with two parameters (Type I and Type II) and the calculation results based on the numerical analysis. As is clear from the figure, the trend between the actual results and the calculation results are almost the same. The cause of the error may be the influence of the play existing in the actual machine, the assumption that the elastic coefficient of each element constituting the actual machine changes with the load, but is assumed to be constant in the numerical analysis. From the above results, it can be said that the stiffness analysis method according to the present invention can be sufficiently used as a design index of the parallel mechanism.
[0037]
Next, the modeling of the elastic coefficient related to the rotating shaft of the bearing as described above will be examined. If the bearing is used for a passive shaft, Φ will be infinite. However, in order to perform numerical calculations, Φ cannot be treated as infinite. Therefore, it is assumed here that Φ is sufficiently larger than other elements of the compliance matrix. This Φ will be considered below.
[0038]
Each element of the compliance matrix in the bearings 1, 2, 3, 4 employed in the improved delta mechanism of the haptic interface is
[Equation 19]

[Expression 20]

It is. Here, the unit of the element for translation is [mm / N], and the unit of the element for rotation is [rad / Nmm]. Further, since the first bearings 12 and 12 in FIG. 4 are connected to the drive shaft of the motor, the value of Φ is assumed to be zero.
[0039]
FIG. 10 shows the change of each element constituting the hand compliance matrix in Type II by changing the value of Φ. The symbol in the figure is
[Expression 21]

Corresponds to each element indicated by.
[0040]
All the elements are stable values in the region of Φ <1 × 10 ⁻⁷ and 1 × 10 ⁻² <Φ <1 × 10 ⁶ . Φ <1 × 10 ⁻⁷ is considered to be equivalent to a state where the rotation axis is fixed, and 1 × 10 ⁻² <Φ <1 × 10 ⁶ is equivalent to a state where the rotation axis is not fixed. Here, assuming that the value of Φ is 1 × 10 ⁵ , this value is sufficiently larger than the other elements and represents the elastic coefficient of the passive shaft.
[0041]
As described above, the parallel mechanism generally has high accuracy, high speed, and high rigidity. However, slight deformation due to elastic deformation of each member or bearing greatly affects the rigidity of the hand. In addition, the rigidity of the hand is greatly affected by the size and location of each member and bearing.
[0042]
Therefore, the present invention individually models the elastic deformation of each member including the bearing constituting the parallel mechanism, and proposes a structural rigidity analysis method based on these models. This rigidity analysis is a method that models the components of the mechanism including the bearings individually and then integrates the components to model the parallel mechanism, so it depends on the individual mechanisms. Absent. Therefore, it can be applied to many elastically deforming mechanisms regardless of serial manipulators and parallel manipulators, and can be made highly versatile, and is particularly effective for parallel manipulators that do not have a specific analysis method for rigidity. It is.
[0043]
Furthermore, the results of the calculation based on the stiffness analysis according to the present invention and the actual results using the improved delta mechanism, which is one of the parallel mechanisms of the present invention, as a translation part of the small 6-DOF haptic interface. As a result of the comparative evaluation experiment, it was verified that this analysis method can be used sufficiently as a design index.
[0044]
【The invention's effect】
Since the present invention is configured as described above, it is possible to provide a highly rigid device operating parallel mechanism design method by obtaining an accurate analysis data method in a parallel mechanism used in an apparatus for operating various devices. In addition, a substantially rigid apparatus operating parallel mechanism can be obtained as a haptic interface from the analysis data. In particular, the versatility of the parallel mechanism for equipment operation including highly rigid bearings by modeling the elastic deformation of each member including the bearings constituting the parallel mechanism individually and obtaining the structural rigidity analysis method based on these models In addition, it is possible to obtain a parallel mechanism including a bearing for device operation with a substantially high rigidity from the analysis data.
[Brief description of the drawings]
FIG. 1 is a schematic diagram of a serial mechanism.
FIG. 2 is a schematic diagram of a parallel mechanism.
FIG. 3 is an explanatory diagram when a force and a moment act on the tip of a link.
FIG. 4 is a perspective view of a serial mechanism used in an improved delta mechanism in a parallel interface according to the present invention.
FIG. 5 is a perspective view of an improved delta mechanism in a parallel interface according to the present invention.
FIG. 6 is a plan view for stiffness analysis of a pair of bearings used in the parallel mechanism in the present invention.
FIG. 7 is a schematic diagram for stiffness analysis of a pair of bearings.
FIG. 8 is a graph showing the results under Type I conditions in a comparison experiment between values obtained by applying the present invention to the improved delta mechanism and actual values.
FIG. 9 is a graph showing the results under Type II conditions in a comparison experiment between values obtained by applying the present invention to the improved delta mechanism and actual values.
FIG. 10 is a graph showing changes in each element constituting the hand compliance matrix when the elastic modulus in the rotational direction is changed.
[Explanation of symbols]
DESCRIPTION OF SYMBOLS 10 Board | substrate 11 Board | substrate side attachment member 12 Bearing 13 1st rotating shaft 14 Motor 15 Arm 16 Bearing 17 2nd rotating shaft 18 1st connection support shaft 19 2nd connection support shaft 20 1st link rod 21 2nd link rods 22 and 23 Bearing 24 Third connection support shaft 25 Fourth connection support shafts 26 and 27 Bearing 28 Third rotation shaft 29 Moving plate side fixing member 30 Moving plate 31 Bearing 32 Parallel link mechanism 33 Arm / rod combination

Claims (2)

A plurality of serial mechanisms, in which one end of a mechanism in which a plurality of members are rotatably connected to each other, are rotatably fixed to the base, and the other end is rotatably fixed to the operation unit,
A parallel mechanism for device operation in which the one end portion of each serial mechanism is fixed to the same base, the other end portion is fixed to the same operation portion, and the operation portions are operated by independently driving the serial mechanisms. When designing
When the hand compliance matrices of the t serial mechanisms constituting the parallel mechanism are Csi (i = 1,..., T),
For the parallel mechanism, the hand compliance matrix Cp is

In the design method of the parallel mechanism for device operation calculated by the following formula:
The elastic coefficient in the x- axis direction that is the rotational axis direction of the rotary coupling member supported by two bearings is ka, the elastic coefficient in the y- and z-axis directions that are perpendicular to the rotational axis direction is kr, and the elastic coefficient around the x-axis. Is 1 / Φ, km is the elastic coefficient around the y- and z-axis, and a is the distance between the bearings supporting the rotary connecting member,
For the parallel mechanism, the compliance matrix Cei of each rotary connecting member i is

The formula of
And the serial mechanism is composed of a plurality of elastic bodies, wherein e is an elastic displacement vector of all elastic bodies, θ is each joint angle vector to which each elastic body including the rotation connecting member is connected, and a Jacobian matrix relating to e is Let Je (θ, e) be the tip compliance matrix Cs of the serial mechanism including all elastic bodies.
Cs = Je (θ, 0) CeJe ^T (θ, 0)
From the equation
The tip compliance matrix Csi of the i-th serial mechanism in the t serial mechanisms is
Csi = Jei (θi, 0) CeiJei ^T (θ, 0)
From the equation
A method for setting a parallel mechanism for device operation, wherein each coefficient is determined so that a value of a tip compliance Cp of the parallel mechanism becomes a desired value. A plurality of serial mechanisms, in which one end of a mechanism in which a plurality of members are rotatably connected to each other, are rotatably fixed to the base, and the other end is rotatably fixed to the operation unit,
A parallel mechanism for device operation in which the one end portion of each serial mechanism is fixed to the same base, the other end portion is fixed to the same operation portion, and the operation portions are operated by independently driving the serial mechanisms. When designing
When the hand compliance matrices of the t serial mechanisms constituting the parallel mechanism are Csi (i = 1,..., T),
For the parallel mechanism, the hand compliance matrix Cp is

In the design method of the parallel mechanism for device operation calculated by the following formula:
The compliance matrices in the first bearing, the second bearing, the third bearing, and the fourth bearing, which are rotational coupling members, are C e 1, C e 2, C e 3, and C e 4, respectively, and are supported by two bearings. The elastic coefficient in the x- axis direction that is the rotation axis direction of the rotary connecting member is ka, y is the direction perpendicular to the rotation axis direction , kr is the elastic coefficient in the z-axis direction, and the elastic coefficient around the x-axis is 1 / Φ, y, When the elastic coefficient around the z-axis is km and the distance between the bearings supporting the rotary connecting member is a,
The compliance matrix for each bearing

The formula of
And the serial mechanism is composed of a plurality of elastic bodies, wherein e is an elastic displacement vector of all elastic bodies, θ is each joint angle vector to which each elastic body including the rotation connecting member is connected, and a Jacobian matrix relating to e is Let Je (θ, e) be the tip compliance matrix Cs of the serial mechanism including all elastic bodies.
Cs = Je (θ, 0) CeJe ^T (θ, 0)
From the equation
The tip compliance matrix Csi of the i-th serial mechanism in the t serial mechanisms is
Csi = Jei (θi, 0) CeiJei ^T (θ, 0)
From the equation
Obtain the value of the compliance matrix for each bearing by putting the actual bearing values into the equation ,
Each coefficient is determined so that the value of hand compliance Cp of the parallel mechanism becomes a desired value,
And Φ of all bearings,
1 × 10 ⁻² <Φ <1 × 10 ⁶ (rad / Nmm)
A method for designing a parallel mechanism for device operation, characterized in that the region is selected in the above-mentioned region.

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