Bi-mobile Mechanisms Designed by Means of Their Inverse Models

Transcription

1 International Conference on Applied Mechanics, Electronics and Mechatronics Engineering (AMEME ) ISBN: ---- Bi-mobile Mechanisms Designed by Means of ir Inverse Models Adriana COMANESCU * and Claudiu ZALESCHI Dept. ory of Mechanisms and Robots, University POLITEHNICA of Bucharest, Splaiul Independentei no., Bucharest, Romania *Corresponding author Keywords: Bi-mobile mechanism, Inverse structural model, Mobile robot, Leg mechanism. Abstract. One the main objective of the mechanism and machine theory is to create new optimal systems. In industrial applications or in robotics the bi-mobile mechanisms are usually used to describe any variable planar curve. structural synthesis of such mechanisms involves new aspects related to their inverse structural models. se models are characterized by zero degree of mobility so that they are of passive modular groups. paper puts into evidence the bi-mobile mechanisms solutions for which the inverse models contain exclusively passive modular groups with for s. Introduction Bi-mobile mechanisms are generally used when it is necessary to be described any curve by a point belonging to the. Consequently they are applied for arm or leg mechanisms in robotics. For their structural modeling the ages with five degrees of freedom and two degrees of mobility may be used [,,,,,, ]. In this purpose one may have in view a single five age with one loop, four ages with two loops and seven s and forty ages with three loops and nine s [, ]. All these ages are included in the classical literature of mechanism theory [, ]. It is also necessary to mention the ages with four degrees of freedom (respectively one degree of mobility). It is important to note the existence of a single four bar and one loop age, two ages (Watt and Stephenson) with two loops and six elements and sixteen ages with three loops and eight elements [,, ]. A general structural synthesis method for the bi-mobile mechanisms previously elaborated [, ] is applied in this paper for the mechanisms with the inverse structural model including the passive modular groups with four s obtained from the Baranov truss with five elements (Figure ). a. b. c. Figure. a) Baranov truss with five s, b) passive modular group PMG, c) passive modular group PMG. When it is necessary to place the extremity of the in a given point this action is equivalent to connect that point through a lower pair by the mechanism. Consequently the initial degree of mobility equal to two unites becomes instantaneously zero due to the introduced lower pair with two constraints. In the selection of the and the it is necessary to emphases the following fundamental aspects: the can not be adjacent to, the same can not be in the same for contour or in the same Watt or Stephenson structure [, ]. inverse structural models may have relative to the the following passive modular group with +++, +, +, +, ++, ++, ++ or s. In the last case the inverse model becomes one of Baranov truss with s and four independent contours [, ]. Some of these aspects are previously analyzed in our papers. By analyzing from this point of view all forty ages [, ] with two degrees of mobility and three

2 loops there are only found some of them for which their inverse structural models are composed by passive modular groups with four elements (Figure ). Structural Characteristics for Linkages with the Inverse Model with Four Link Passive Groups Having in view their passive modular group for their inverse structural model only some of forty ages (Figure, Figure, Figure, Figure, Figure, Figure, Figure, Figure ) with three independent contours are characterized by four passive modular groups in their inverse model (Figure ). se ages may be nominated by the following parameters: exterior loop class/ each inner loop class. Figure. --- Figure. --- Figure. --- Figure. --- Figure. --- Figure. --- Figure. --- Figure. --- and effector solution determined by its inverse model as previously mentioned for the --- age (Figure ) may be synthesized in the -effector matrix (Eq. ) A[i,j] =. () In this symmetrical matrix with the zero element for its principal diagonal the is i element and the effector the out-put is j. So that one may find distinct solutions for the and effector, but only two ones have their inverse structural models with four element passive groups (Table ). --- age (Figure ) has some particularities. It contains two symmetrical s named and and two four element ages the,,, age and the,,, -effector matrix given by Eq. puts into evidence solutions, but only of them are distinct ones (Table ). For the --- age (Figure ) one may find the solutions mentioned in Table for bases and effectors in the previously mentioned conditions.

3 A[i,j] =. () Table. inverse structure models for the --- inverse structure model modular passive group GMP(,,.)+GMP(,,,) GMP(,,,)+GMP(,,,) Table. effector distinct solutions for the --- Basis i and effector j A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] Table. effector distinct solutions for the --- Basis i and effector j A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] Table. inverse structure models for the --- modular passive group GMP(,,.)+GMP(,,,) GMP(,,,)+GMP(,,,) GMP(,,.)+GMP(,,,) GMP(,,,)+GMP(,,,) Table. inverse structure models for the --- modular passive group GMP(,,.)+GMP(,,,) GMP(,,,)+GMP(,,,)

4 --- age (Figure ) has the exterior contour similar to the inner loop so that the number of distinct solutions is (Table ) from ones nominated in the - matrix (Eq.). A[i,j] =. () solutions for the and effector with its particular inverse structural models are given in Table. --- age (Figure ) has the exterior loop structurally identical to the inner contour and the following symmetric s ===, =, =. By using the matrix (Eq.) with non-zero elements specifically to this age, one may distinguish only distinct solutions mentioned in Table. required solutions with their passive groups configuration are given in Table. A[i,j] =. () Table. effector distinct solutions for the --- A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(, ) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) Table. inverse structure models for the --- modular passive group GMP(,,,)+GMP(,,,) GMP(,,,)+GMP(,,,) Table. inverse structure models for the --- modular passive group GMP(,,.)+GMP(,,,) GMP(,,,)+GMP(,,,) Table. effector distinct solutions for the --- A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,)

5 Through a similar matrix solutions are obtained for the bases and the s for the --- age (Figure ), but only two of them (Table ) satisfy the initial imposed conditions. --- age (Figure ) has the exterior loop similar to the inner one and the following symmetrical s ; ;. distinct solutions are presented in Table and the desired configurations are also shown in Table. --- age (Figure ) has the, symmetrical s and two contours ---- and ---- are similar. Its distinct solutions are presented in Table and the solutions for design in Table. As to the --- (Figure ) the relevant information about the structural synthesis of the bi-mobile mechanisms are shown in Table and Table. One may also note the Stephenson structure given by Table. inverse structure models for the --- age Basis Output modular passive group GMP(,,,)+GMP(,,,) GMP(,,,)+GMP(,,,) Table. effector distinct solutions for the --- A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(,) A(, ) A(,) A(,) A(,) A(,) A(,) Table. inverse structure models for the --- Basis Output modular passive group GMP(,,,)+GMP(,,,) GMP(,,,)+GMP(,,,) Table. effector distinct solutions for the --- A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] A[,] Table. inverse structure models for the --- Basis Output modular passive group GMP(,,,)+GMP(,,,) GMP(,,,)+GMP(,,,) GMP(,,,)+GMP(,,,) GMP(,,,)+GMP(,,,) Bi-mobile Mechanism Designed by Using the Inverse Model with Four Link Passive Groups For example a new solution for a bi-mobile mechanism may be design by using the --- age (Figure ). A structural solution for the and the must be selected from Table, so that a certain point of the effector is able to describe any curve. When one of the solutions is adopted from Table, the inverse structural model of the bi-mobile mechanism contains only passive with four elements.

6 Figure. bi-mobile mechanism with the and the. Figure. bi-mobile mechanism with the and the. In Figure the mechanism has consequently the and the. By using the --- age (Figure ) and Table, the designed bi-mobile mechanism from Figure, in which the is and the is achieved when the solution is adopted from Table. Summary paper brings in attention the structural synthesis method for bi-mobile mechanisms. This makes possible to create numerous absolutely new patentable solutions for various robotic or non-robotic systems. In some previous papers [, ] various solutions are shown having for their inverse structural models different passive modular groups. This paper reveals those solutions from the ages with five degrees of freedom and three independent loops, which have for their inverse structural models only passive modular groups with four elements. Eight ages of them (Figure, Figure, Figure, Figure, Figure, Figure, Figure, Figure ) satisfy the previously mentioned conditions, from which one may find solutions for the and the (Table, Table, Table, Table, Table, Table, Table, Table ). re also presented two solutions achieved in a special environment (Figure, Figure ). placement of the active pairs is the next problem for creation of new bi-mobile systems. References [] Chr. Pelecudi, Bases of Mechanisms Analysis (Bazele analizei mecanismelor), (Editura Academiei Române), Publishing House of the Romanian Academy, Bucharest,. [] F.R.E. Crossley, Structural Synthesis of a Four Bit Binary AddingMechanisms, -th ASME Mechanisms Conference, Paper no. MECH, Atlanta - Oct.,. [] N. Manolescu, Fr. Kovacs, A. Orănescu, Teoria mecanismelor şi a maşinilor, Editura Didactică şi Pedagogică, Bucureşti,. [] I.I. Artobolevski, Théorie des mécanismes et des machines, Editions Mir, Moskva,. [] N. Rojas, F. Thomas, On closed-form solutions to the position analysis of Baranov trusses, Mechanism and Machine ory, vol., pp.-,. [] J. Angeles, Fundamentals of Robotic Mechanical Systems: ory, Methods and Algorithms, Springer-Verlag New York,. [] Adr. Comănescu, D. Comănescu, I. Dugăeşescu, L Ungureanu, Optimal inverse models for bi-mobile mechanisms of walking robot legs, DAAAMinternational Scientific Book, vol., pp.-, Publisher DAAAM International Vienna, ISBN ----,.

7 [] Adr. Comănescu, D. Comănescu, I. Dugăeşescu, L Ungureanu, C. Alionte, Classical modules applied to the mechanisms structural analysis and synthesis, th World Congress in Mechanism and Machine Science, Taipei, Taiwan, - October,. [] M. Plecnik, J.M. McCarthy, Design of Stephenson Linkages that guide a point along a specified trajectory, preprint submitted to Mechanism and Machine ory, April,.