On Environment's Mathematical Model and Multilegged Walking robot Stable Evolution

Transcription

1 Research Article Special Issue On Environment's Mathematical Model and Multilegged Walking robot Stable Evolution Marcel Migdalovici, Sergiu Cononovici, Victor Vladareanu*, Gabriela Vladeanu Robotics and Mechatronics Department Romanian Academy, Institute of Solid Mechanics, Bucharest, Romania; * Mihai Stelian Munteanu Faculty of Electrical Engineering, Technical University of Cluj-Napoca Daniela Baran Dynamical Systems, National Institute for Aircraft Research "Elie Carafoli", Bucharest, Romania; Hongbo Wang, Yongfei Feng Parallel Robot and Mechatronic System Laboratory, Yanshan University, Qinhuangdao, China yf_feng@126.com Published online: 5 March 2018 Abstract: Mathematical modeling of the environment, using our study from previous papers on the stability of the dynamical systems evolution, in the general case of the dynamical systems that depend on parameters, which approach the phenomena from the reality, is proposed and justified in the first part of the paper. The human perception of the reality is characterized by the quality of the continuity perception for the functions that define any concrete dynamical system from the reality and also by possible finite number of such functions discontinuities. The property of piecewise continuity for the functions that define the dynamical system that approaches environment's phenomenon implies the property of stable regions separation in the free parameters domain of the dynamical system. The existence of the stable regions in the free parameters domain permits to optimize the evolution of the dynamical system using a compatible criterion for optimization, in other words, we can realize a stability control of the dynamical system evolution. We remark that all dynamical systems that depend on parameters from the literature have the property of separation between stable and unstable regions in the free parameters domain. In the second part of the paper our theory on the stability control of the dynamical systems from the environment is applied here for specified multi-legged walking robot model that depend on parameters in two dimensional and three dimensional cases of legs evolution. Our critical position theory on the walking robot evolution is reminded and developed on our analyzed cases of walking robot models inspired from the environment. Keywords: environment's mathematical model, environment's dynamical system, stable region, multi-legged walking robot, critical position, kinematics analyze. I. INTRODUCTION Any dynamical system from the environment can be considered as dynamical system in terms of its defining parameters without fixing their values as physical parameters (in particular mechanical parameters), geometrical parameters, possible chemical, economical, biological parameters or other. The matrix that defines the linear dynamic system has the matrix components assumed to be with real values, and the matrices that intervene in the exposure of the method are also with components real values. In the last papers we assumed that the matrices from the mathematical model have the complex values such that the real values are also taken into account as particular case of complex values. This hypothesis assures a new method of analysis, in the complex domain, on the stability of the dynamical system. Our theory on the stability control of the dynamical systems is applied here for specified walking robot models that depend on parameters. The critical position of the walking robot evolution is reminded and analyzed on some cases of the walking robot leg in kinematics analyze [1-11, 13, 22-33]. An important idea that arises from all concrete dynamical systems that depend on parameters from the literature is on the property of separation between stable and unstable regions in a specified domain of free parameters [ , 21]. The stable and unstable regions are separated in the domain of free parameters by a boundary. The property of separation can be seen by the fact that the stable and unstable regions are open sets, excepting the points on boundary. This separation aspect creates the freedom of stability control on a neighborhood of any stable point in the open stable region of the dynamic system. We discovered mathematical conditions of the stability regions existence for the dynamic systems that are inspired from the environment, using various results from applied mathematics domain [14, 16, 17-21]. In this paper we define the environment mathematical model analyzing some aspects on the conditions of separation between stable and unstable regions in the free parameters domain of the linear or nonlinear dynamical systems, calling the QR algorithm applied on the real matrix that defines the doi: J Fundam Appl Sci. 2018, 10(4S),

2 linear dynamic system or on the "first approximation" of the nonlinear dynamic system, using operations in the complex domain by the matrix "shift of origin". The dependence of this matrix eigenvalues on the matrix components properties intervene in justification of separation property. The components of the real matrix that define the linear dynamical system depending on parameters are assumed to be piecewise continuous in the free parameters and analyzed to contribute for the final argumentation of the environment model. II. MULTI-LEGGED WALKING ROBOT MODELS In the following we analyze some problems of walking robot evolution by synchronization of its legs evolution in kinematics hypothesis. Extension to dynamical evolution of walking robot, on particular case, calling the above theory is intuitively accepted and is very attractive to be studied. The physical and mathematical walking robot model, with physical model consists from a platform and six similar legs that have a joined extremity attached to the robot platform and a synchronized evolution in the vertical plane are firstly studied (fig.i). Another case studied here, based and inspired on the initial two dimensional evolution technique used for performing leg evolution, consists in three dimensional evolution of multilegged walking robot each leg, exposed in fig. 5 and explained here. Each of the two dimensional leg evolution take into account a leg compounded from two jointed segments and with this joint named knee joint, describing a circle arc route in an evolution of the leg and with the base extremity of the leg describing an imposed route on the ellipse arc (fig.2). The leg base point in its evolution is assumed having constant speed component on horizontal direction, hypothesis that imposes the concrete leg evolution of the robot, excepting the of the knee joint, notion introduced by the authors and detailed also in the following. The physical model of the walking robot leg, in two dimensional case of the legs evolution, is described in fig. 2. The similar physical model is used for three dimensional evolutions of legs in multi-legged walking robot model. The points that specify the extremities of the continuous domain on circle arc where the knee joint is moving, in two dimensional evolutions of the legs, have such positions where the inferior segment of the leg is normally on the ellipse arc. The physical model used for three dimensional legs evolution of the multi-legged walking robot emphasizes also possible critical points existence. The critical points for three dimensional legs evolution, if they exist, will specify the extremities of the continuous domain where the knee joint is moving. The point denoted by O C is joint extremity of the leg attached to robot platform and the segments denoted by O C P and PQ having the length a respectively c are the components of the robot leg. The point P is the knee joint of the leg and the point Q is the leg base point. Fig.I Walking robot physical model. evolution in the neighborhood of possible critical positions analyzed below. The physical model and attached mathematical model of the robot leg in the first case analyzed assure the possibility to identify the existence of the critical position on the circle arc Fig. 2. Physical model of walking robot leg. The knee joint P describes, in cyclic movement of the robot leg, a route on circle arc, and a base point Q of the leg an imposed elliptic route on the superior ellipse arc Q B Q A. The circular route is considered with the radius a and centre O C while the elliptic route is assumed with the semiaxes a, b and centre O E. The elliptic and respectively circular routes are transverse involving impose constant speed component of the leg base point, on horizontal direction excepting a neighborhood around the critical position. The critical point of the robot leg evolution is the point P c on the circular arc and corresponding Q c point on the elliptic arc where the movement direction of the "knee joint" is changed. The orthogonal system of coordinates is identified for the physical model from fig.2 by the following values of the coordinates, defining the figure points: J Fundam Appl Sci. 2018, 10(4S),

3 O C (a, h), O E (a, b 1 ), P( x P, y P ), The linear evolution of the variable x between O and 2a Q( x Q, y Q ), Q A ( x A, O), Q B ( x B, O). is assumed as below, where the constant speed and initial condition x O are considered: The following conditions are used on the parameters of the x(t ) t x leg mathematical model: O (6) a b b 1 O; a c h In critical position of the leg "knee joint" evolution and in corresponding leg "base point" position is necessary null speed, important property that will be used in concrete robot leg mathematical model description. The points P * and P * in the figure 2 delimit the maximal domain on the circular arc where the point P can be moved The explicit functions deduced from (1) have the form: 2 y P h (2ax x ) 1/ 2 ; ( x 2 P x Q ) 2 ( y P y Q ) c 2 O is imposed. C C C P (a x) [h b 1 (b l a)(2 ax x ) ] (5) a 2 c 2 2ac cos An evolution cycle of the robot leg can start from the point Q B, moving on the superior ellipse arc up to point Q A, using an evolution law on horizontal axis defined by (6), excepting a neighborhood of critical points. The covering domain of variables is reminded below. x [O, 2a], y P [h a, h a], y Q [O, b b 1 ] (7) A B The explicit functions selected for the physical models from figs. 3 and 4, are as: because in this position the segments P * Q * and P * Q * are A A B B y h (2ax x 2 ) 1 / 2 P normally on the ellipse arc and the distance up to ellipse arc is (8) y b / a (2ax x 2 ) 1 / 2 b Q 1 minimum in the neighbourhood. This property of separation between the regions of existence (implicit stability) and In first concrete case of walking robot leg physical model inexistence (instability) emphasized by analyzing of the knee the below relation is assured, where x [ x QA, x QB ] : joint P kinematics evolution from the environment. (a x) 2 [(b / a)(2ax x 2 ) 1/ 2 The mathematical model of the robot leg corresponding to physical model from fig. 2 is described by the relations: (b h)] 2 a 2 c 2 2ac cos (9) (x a) 2 ( y h) 2 a 2 O. ; (1) Because the knee joint speed is natural of zero value in critical positions where the sense of motion is changed, using ( x a) / a ( y Q b 1 ) / b 1. also Cauchy conditions of continuity is necessary in A covering domain for the variables from (1) is defined neighborhood of critical points to assure continue values of below: displacement and speed as functions on time. x [O, 2a], y P [h a, h a), y Q [O, b b 1 ]. (2) In the following we describe particular cases of walking robot legs using physical models from the figs. 3 and 4. The walking robot's physical model is defined with the help of the Let P( x P, y P ) (3) corresponding robot leg definition. For the physical robot leg model from fig. 3 we see the 2 1/ 2 y Q b / a (2ax x ) b 1 (3') pivot point O P of the leg, the mobile joint P of the leg that be the point on the circle arc and Q( x Q, y Q ) the point on the ellipse arc that corresponds for one position of the robot leg evolution. The condition The segment O C Q in the triangle O C PQ can be expressed as function of the angle (O C PQ) denoted : (O Q) 2 (O P) 2 (PQ) 2 2(O P)(PQ) cos (4) The following analytical relation is formulated, where the variable x Q is assumed an independent variable and is denoted by x : 2 2 1/ 2 2 The relation (5) is used to evaluate the angle (O C PQ) as function of variable x, for x [O, 2a]. The points Q A ( x A, O) and Q B ( x B, O) have the abscises 2 1/ 2 2 1/ 2 x A a a (1 (b 1 / b) ), x B a a (1 (b 1 / b) ), with x A, x B being strictly between O and 2a. describes a circle arc in the leg's evolution on cycle and the point Q of the robot leg base positioned in our case on an ellipse arc of centre O Q and semi axes length a and b. The orthogonal system of coordinate axes is similar as for fig. 2. Fg.3. One physical model for walking robot leg. J Fundam Appl Sci. 2018, 10(4S),

4 Let O P (a, h), O Q (a, b 1 ), Q A ( x A, O), Q B ( x B, be points O) from figs. 3 and 4. An evolution cycle of the leg base point from fig.3 is defined by the successive evolution Q A, Q 1, Q B, on the horizontal axis followed by the succession Q 2, Q 3, Q A on the superior ellipse arc. The critical points are not identified on the physical model from fig.3 but there exist on the physical model from fig.4. These two models are different through the value of parameter h, in the first case the value h 1 and in the second case the value h 2 h 1 that imply the corresponding one such position. modification of the parameter b 1. The conditions a b b 1 O; a c h are assumed, where the length of the inferior component of the robot leg is c. For the physical model from fig.3 is assumed c a. The analytical method for identification of critical points on the circle arc and implicit the critical points on the ellipse arc, for concrete physical model from fig.4, is deduced by solving the equation that depends on x Q imposed by the condition of segment P c Q c orthogonal position. A three dimensional leg evolution physical model for multi-legged walking robots is described in fig. 5, [15]. In the critical point is necessary to be assured zero value of the leg base point speed. We remark that the property of walking robot stability (existence) position is also maintained in a neighborhood for The leg is compounded from superior component BtQt defined by the extremities points Bt, Qt jointed in pivot point Bt attached to the body of the robot and inferior component QtPt with "knee joint" Qt and base point Pt. For the length of components BtQt and QtPt has assumed a constant value. The base point Pt is moving on the ellipse arc between points P I and P F, in vertical plane, orthogonal on axis Oy, using uniform accelerated displacement on the horizontal direction up to the median point P M, on the ellipse arc, and symmetric displacement assured up to the final point P F. Fig.4. Other physical model of walking robot leg. Our critical position definition in the robot leg evolution, reminded here by us, is the position where the point P c on the circular arc and corresponding Q c point on the elliptic arc have the property of changing the movement direction of the "knee joint". In the critical position of leg "knee joint" evolution and in the corresponding leg "base point" evolution we remark the necessary null speed of leg "knee joint" and we mention again that this is an important property used in assurance of concrete evolution of the robot leg. The critical points P c and Q c, in our concrete case, define the extremities of one maximal domain on the circular arc where the point P can be moved and in this position the Fig. 5. Physical model of three dimensional walking robot leg. segment P c Q c is normally on the ellipse arc. The sense of movement is changed on the circle arc in the extremities of the The joint point Bt attached to the body of the robot is continuous domain of evolution for "knee joint". This notion moving simultaneously, having linear route parallel to the on critical point can be generalized in the case more general of walking robot movement where the ellipse arc can be axis Ox, using uniform displacement between initial point B I substituted by another continuous curve or in three up to median point B M and symmetric displacement assured dimensional legs evolution that is described as follows. between the median point B M up to final point B F. J Fundam Appl Sci. 2018, 10(4S),

5 The trajectory of the "knee joint" point Qt, unique identified in the vertical plane, defined by the points The coordinates for the points Bt, Pt are identified in function on time as below: Bt, Qt, Pt, in each time t, having defined the distance x Bt x B I v B t, y Bt 0., z Bt 1.5L between its, with the length of segment BtPt dependent on x a t 2 / 2, y 1.5L, (13) Pt P Pt time t, is studied for possible critical points identification, similar as in two dimensional cases described above. The z (b / a)(a 2 ( x d / 2) 2 ) 1 / 2 b following formulas, calling the geometric data from fig. 5 and physical data, are used. Pt Pt 1 In the above relations 0 t t M, a, b are ellipse semiaxes and 0 < b 1 b. Coordinate z is imposed negative for the The three dimensional trajectory of the point Qt is ellipse centres from the fig. 5 so that z Pt 0, b b 1. orthogonal projected on the plane xoz using the projected point Qt BtPt 0 from this plane. The critical position of the The length of the segment is calculated from: point Qt, in its three dimensional evolution, if there exists, is / 2 BtPt (( x Pt x Bt ) ( y Pt y Bt ) ( z Pt z Bt ) ) (14) identified by critical position on the plane projected trajectory of the point Qt 0 where the direction of movement must to be The angles from the vertical triangle BtQtPt are changed. The point Qt calculated, using the lengths of the triangle sides, by the 0 is orthogonal projection of the relations of type: point Qt on the side Q Bt Q Pt, which is parallel with the axis Ox, and where the triangles QtQ (BtQt) 2 (BtPt) 2 (QtPt) 2 Bt Q Pt, Q 0 B 0 P 0 have the cos( (QtBtPt))= (15) sides respectively parallel. 2BtQt BtPt The uniform accelerated displacement of the point The angles values of the vertical triangle BtQtPt from the denoted P, on the horizontal line, from the point P I up to physical model described on fig.5 permit to evaluate all middle of the segment P I P F, is described by the relation: necessary angles or sides from the physical model. The critical point where the direction of movement is changed for the x (t) a t 2 / 2 (10) projected point Qt p 0 is searched for the parameter time in the p The parameter a interval [0, t M ). p is constant acceleration in the displacement of the horizontal coordinate of the point denoted P. The mathematical model of the three dimensional legs The uniform displacement of the point Bt, on parallel line evolution for multi-legged walking robot, explained above, with axis Ox, from the point B I up to middle of the permits to identify possible existence of the critical position segment B I B F, is described by the relation: for knee joint point, calling specialized computer program and this is planed in the next paper. x Bt (t) v B t (11) The parameter v B is constant speed in the displacement of the point Bt on the parallel line with axis Ox. The parameter L is introduced for correlating some data from the three dimensional mathematical model of walking robot leg. The following lengths are defined BtQt =2L, QtPt 3L, BtB 0 1.5L, PtPt 0 1.5L. The walking robot with six legs is considered, so that to close one cycle of evolution, such that the following relation arises P I P F 6 B I B F. This relation is justified by the hypothesis on the separation and successive displacement of each leg in cycling evolution. The necessary time for simultaneously arriving in the middle of the routes B I B F, P I P F, by point Bt respectively Pt, is denoted by t M. The following relations are true: v B t M =d /12, v B =d / t M /12; a t 2 / 2 d / 2, a d / t 2 p M p M III. CONCLUSIONS A mathematical characterization of the environment and the application on the walking robots dynamical system stability control, the walking robot being considered as particular case of a dynamical (kinematical) system that depends on parameters, is performed. The mathematical conditions on the separation between stable and unstable regions, in the domain of the dynamical system free parameters, discovered by us, are emphasized, calling the algebraic operations in complex domain that permit new results in the stability theory. These conditions of separation represent the conditions that describe the dynamical system from the environment. The property of separation described in the paper, encountered also in many other dynamical systems from the literature, where only are contemplated without mathematical justification, is very important property because it creates the possibility of the stability control of a dynamical system from (12) the environment by optimization of the system evolution using the compatible criterion in the stability regions of the free parameters domain. The study for the dynamic systems can be performed, with conservation of the fundamental idea, and for J Fundam Appl Sci. 2018, 10(4S),

6 kinematics of the walking robots. The property of stable region separation is described in our cases of walking robot leg with the possibility to be extended on other cases. The critical position on the particular case of the walking robot is defined in the paper and a mathematical method of its identification is performed. We remark that our study has not exhausted the problem on environment mathematical characterization and the problem on dynamical systems stability control or on kinematical stability control however an interesting domain from the environment has been opened. ACKNOWLEDGMENT This work was developed with the support of MEN- UEFISCDI, PN-II-PT-PCCA , VIPRO project No.009/2014, Romanian Academy and "Joint Laboratory of Intelligent Rehabilitation Robot" collaborative research agreement between Yanshan University, China and Romanian Academy by IMSAR, RO by China Science and Technical Assistance Project for Developing Countries (KY ), REFERENCES [1] Hirsch, M. 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