THEORETICAL AND EXPERIMENTAL RESEARCHES ON THE DYNAMICS OF MOBILE ROBOTS

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1 Revista Mecanisme şi Manipulatoare, Vol., Nr., 6, pag ARoMM-IFoMM HEOREICAL AND EXPERIMENAL RESEARCHES ON HE DYNAMICS OF MOBILE ROBOS Nicolae DUMIRU *, Mirela CHERCIU** *) Prof. dr. ing. Universitatea din Craiova, *) Conf. dr. ing. Universitatea din Craiova, Abstract: his paper presents a package of procedures for the kinematic and dynamic study of a multimobile hexapode robot. he dynamic and kinematic procedures are based on a matrix formalism that enables the rapid computer implementation, both graphical and computational. he kinematic procedures are used to solve two important problems: -the identification of the generalised co-ordinates by eliminating the restrictions and the redundant mobilities ; - the establishing of the variation laws for different kinematic parameters. he dynamic procedures, based on the Newton-Euler formalism, are useful to determine the connecting forces in the kinematic pairs, necessary further on, to the processing of the inverse dynamic analysis. 1. Introduction In [3] and [6] there were studied some aspects regarding the structure and the kinematics of a hexapode walking robot. he working out of the structural and kinematical models had as source of inspiration the movement from the living world, namely the movement of the hexapodous insects from the Blatodea order.. he structural models he structural model of the robot which precisely comply with the structure of the insect s biomechanism, includes 39 kinematical elements, 4 monomobile kinematical pairs and 14 three-mobile kinematical pairs. Due to the model's complexity regarding the kinematical functions, it was proposed a structural systematization based on two criteria, as follows: 1) he adjustment of the structural model for the type of locomotion having in view; ) he reappraisal of the rolling movement tarsus-ground by the tarsus ground contact. he structural models have been thought out for four types of locomotion: - he displacement on horizontal ground following a straight line; - he displacement on horizontal ground following a curve line; - he displacement on horizontal ground with the avoidance of some obstacles; - he displacement in conditions of normal walking, race or with rapid turnings, inclusively the displacement on rough ground. he calculus for the freedom degree is detailed in [3]. Fig. 1 he general structural model Fig. he general kinematical model 67

2 For the situation when the contact with the ground, modeled through superior pairs (fourth class), the freedom degree for the most general structural model in Fig. 1 is M=6. 3. he kinematical modeling he kinematical models for the four types of locomotion are presented in Fig.. he mathematical models developed for the identification of the kinematical parameters are presented in detail in [3] and [6]. 4. he dynamical modeling In order to find the dynamical models it was used the Newton - Euler algorithm, made up with the method of Lagrange multipliers heoretical considerations Achievements in the field of the mathematical and mechanical modeling of the locomotion through stepping: -in [1], it is shown that this type of displacement is used to move a body in a referenced, overall system by controlling the relative movement of the legs reported to the body. - in [1] it is also shown that the stepping sequence begins with the acceleration of the body in a constant mode, then it is a period with constant velocity of the body and finally, the body's velocity decreases in a constant mode till reaches zero value. - in [] it is having in view the body's displacement only on horizontal, straight line, considering the kinematical and dynamical parameters (the load factor, the velocity and the acceleration of the body) to be known. -the variation curve of the body's velocity is considered to be trapezoidal shape, because it is easier to be analyzed and much closer to the practical one in []. 4.. he elaboration of the mathematical models he vector of the generalized coordinates has the form: jt q r,, where: r r1, r,..., r, n n is the number of kinematic elements j 1 3,, - the Euler angles for the case when the spatial orientation is given by these angles. or: q r, p, the vector of the generalized coordinates if the spatial orientation of the kinematic elements is given by the Euler parameters. o define the movement of one kinematic element, there are considered referenced three-axis systems, as follows: '( x ' y ' z ') - solidary to the kinematic element (x,y,z) - the overall system Let it be a kinematic element and a point E whose position reported to the overall system is expressed by the known relationship: r As ' E, (1) where A is the transformation matrix of the coordinates, at the passing from the system ' (x',y',z') to the system (x, y, z). It is known that: A A I, () where I is the unitary matrix. From [1] it is known that A A is an anti-symmetrical matrix. It is written A A m / A (3) A m A (4) here are considered two vectors r 1 and r in the referenced system (x, y, z) and the vectors the system ' ( x', y', z'). 68 ' 1r and ' r in

3 We can write the equations: or 1 1 ' r Ar A (5) r1 ' A r 1A (6) he virtual displacements of the point E can be obtained by differentiating (1), as follows: s ' (7) Based on (5) we write r 1 m and r A E m A s ' (8) r E m s (9) r E m A m' A (1) m A m ' (11) he relationship (4) becomes: A A m A A A m ' (1) or m' A A (13) he virtual displacement for the point E will be: r Am ' s ' E (14) Let F = [F 1... ] be the active forces which act in the points E k of the kinematic elements, k 1, n, and ' = [' 1, '... ] (15) the moments acting upon the kinematic elements, considered in the its own system '(x', y', z'). We derive with respect to time, the equation: AA I A A A A (16) ( A A ) A A A A So A A is an anti-symmetrical matrix. A A A A (17) ' A A A A A A A (18) A', A ' A A ' A ' ' (own system) he movement equation for a kinematic element has the following form: M r F J ' ' ' J ' ' ' A A A A (19) he equation () can be written under the form: r M r F m' J ' ' ' J ' ' ' (1) he mechanism configuration given by the kinematical elements and joints leads to the system of equations, having the form: E( q, t ) () 69 ()

4 or By differentiating these equations we get: E( r,, t ) (') ' ' Jr r Jm m (3) It is known that: j q r, (4) or q r, m, where: m defines the spatial orientation of the kinematical elements, when the Euler parameters are used. he relationships (1) and (3) can be shortly written: q M q Q (5) respectively: where Q is the vector of the generalized forces Jq q (6) Q F, ; A ' (7) According to the theorem of Lagrange multipliers, in analogy with (5) and (6), the movement equation become: M q Q q Jq q (8) M qq Jq q (9) M q Jq Q (3) In order to get the velocities, () is derived with respect to time and it is obtained: E E 1 E Jqq Jr r Jm ' ' b q t t Jq (31) t o get the accelerations (31) are derived with respect to time and it is obtained: E E Jqq J q q Jqq Jqq a (3) t t he equation (3) can be written detailed as follows: 1 E q Jq J q q t ' (33) Jr r Jm ' a (34) aking into account (1) and (34), (3) can be written in detail as follows: M r Jr F J ' ' Jm ' ' ' J ' ' where, the sign "~" was used to note the anti-symmetrical matrices. Combining (34) and (35), the movement equations for a spatial mobile system are: (35) 7

5 Written in a short form, (6) becomes: 4.3. Application. r M J r F J ' J ' ' m ' ' J ' ' Jr Jm' a M J q q Q Jq a (36) (37) Fig. 3 he dynamic model Fig.4 he kinematic model of the left fore-leg he presented mathematical models were necessary in the dynamic study of an experimental hexapode robot whose structure and kinematics had as an inspiration source the hexapodous insects movement. For the dynamic analysis we used as input data the forces that resulted during the contact between each of the six legs and the lead surface and the output data is the distribution of these forces resulted from the characteristics of the stepping process. In Fig.3 it is presented the dynamic model of the movement under general circumstances. Fig. 5 he experimental model Fig. 6 he simulated model For the dynamic analysis, we assume that the geometric parameters and the components of the inertia torques are already known, following the determination of the variation laws of the generalized coordinates and of the liaison forces torques in the kinematic couplings. he computational processing of the data allowed to obtain all the variation laws in time of the kinematic parameters, such as: 71

6 x Vy Vz V (mm/ s) vs. t (s) 1-1 Wx Wy Wz W (deg/ s) vs. t (s) Fig. 7 he variation laws of the velocity s Fig. 8 he variation laws of the angular velocity s components components for the element 6 for the element 6 x y z (N mm) vs. time (s) Fx Fy Fz F (N) vs. time (s) Fig. 9 he variation laws of the moment s components in the A s pair Fig. 1 he variation laws of the connecting force s components F 1 It was built an experimental model in Fig.5 which follows accurately the same structure and geometricalkinematic parameters that characterise the stepping process of a hexapodous insect. 5. Conclusions he paper presents an intricate mathematical formalism useful in the dynamic study of the mobile robots. his formalism is based on the Newton Euler method, completed by the Lagrange multipliers and adapted to the complex study of the mobile spatial systems. he hexapode robot that constitutes the object of this paper was realised through the structural, kinematic and dynamic study of a hexapodous insect, Blatodea zoological order. he processing of the dynamic models on a computer and the three dimensional simulation of its behaviour offers the necessary information to proceed with the dynamic study on the experimental model. References [1] Baek, Y.S., Foot placement during legged locomotion, Eight World Congress on the heory of Machines and Mechanisms, Prague, Czechoslovakia, [] Baek, Y.S., E.F. Fichter and B.L. Fichter, Legged Locomotion at Constant Body Velocity, Robotics and Manufacturing Recent rends in Research Education and Applications, ASME Press, 3,199. [3] Dumitru, N., Mecanisme Spatiale. Modelare, cinematica si dinamica prin metode computerizate, Editura Universitaria, Craiova.. [4] Dumitru,N., Modeling some hexapode mechanisms inspired from insects, Ninth World Congress on the heory of Machines and Mechanisms, Milano, Italy, [5] Dumitru, N., Considerations upon the structural-kinematic modelling of a multi-motion hexapode robot, YUCAM VRNJACKA BANJA 97 [6] Dumitru, N.,Ciupitu, I., heoretical and Experimental Contributions to the Modeling of the Biological Robots, 8-th International Workshop on Robotics in ALPE-ADRIA-DANUBE REGION, RAAD'99, Munchen, Germany, pag [7] Gorinevsky, D.M., Sneider, A. Yu, Force control of Legged Vehicles over Rigid and Soft Surfaces, he International Journal of Robotics Research, Vol.9, No., April