Matlab Based Interactive Simulation Program for D Multisegment Mechanical Systems Henryk Josiński,, Adam Świtoński,, Karol Jędrasiak, Andrzej Polański,, and Konrad Wojciechowski, Polish-Japanese Institute of Information Technology Aleja Legionów 4-90 Bytom, Poland {hjosinski,aswitonski,kjedrasiak,apolanski,kwojciechowski}@pjwstk.edu.pl Silesian University of Technology Akademicka 6 44-00 Gliwice, Poland {Henryk.Josinski;Adam.Switonski;Andrzej.Polanski; Konrad.Wojciechowski}@polsl.pl Abstract. This paper presents principles of designing multisegment mechanical system. Represented is a model of a single segment, its extension in form of a couple of segments and the final construction a fragmentary and simplified silhouette of a human form named the biped. This paper describes procedure of construction of the biped s digital model using Matlab package. It also discusses test run of a single experiment and algorithm of the calculations realized in the single step of integration. Introduction Scientists create physical and mathematical models of movement of a whole human form or its specified parts. Mathematical models enable creation of digital models to conduct computer simulation. Model testing makes it possible to analyse decomposition of forces and torques in a non-invasive way. The purpose of this research was to build a simple multisegment movement model in order to determine the principles of adding the successive segments and how to avoid their break-up. This is how the silhouette named biped was created. Its digital model created using Matlab package should enable to carry out simulation of the biped s movement. Literature related to this subject is extensive. Analysis of contents of a movement model was included in [9]. The dynamics of planar human body motion, solved with a non-iterative matrix formulation was presented in []. Worth mentioning is also item [3] of the bibliography where Brubaker et al proposed a model based on the Anthropomorphic Walker [5, 6], a physics-based planar model with two straight legs, a single torsional spring and an impulsive collision model. The Anthropomorphic Walker is simple, as it only exhibits human-like gaits on level ground. Brubaker et al [] introduced also the Kneed Walker a complex model of bipedal locomotion based on biomechanical characterizations of human walking [7]. It has a torso and two legs with knees and ankles. It is capable of exhibiting a wide range of plausible gait styles. A mathematical model of the swing phase of walking was presented in [8].
H. Josiński et al. Fig.. Decomposition of forces for a single segment Single Segment Movement Model Let s consider single segment with the mass m and length l, where centre of the mass is positioned in the point (X,Y ), its ends are at the points (x,y ),(x,y ) and the angle of vertical inclination is ϕ (Fig. ). Let s presume action of external forces: F with components F x and F y, F with components F x and F y and external control torque M. The following equations describe the segment s state: Ẍ = F x + F x m Ÿ = F y + F y m () g () ϕ = l [(F x F x ) cos ϕ + (F y F y )sin ϕ] I + M I where g represents acceleration of gravity and I is segment s moment of inertia relative to its axis of rotation crossing centre of the mass I = ml /. Equations allow to determine acceleration s vertical and horizontal component and also angular acceleration. State variables are represented by the centre of the mass coordinates ) (X,Y ), vertical and horizontal components of centre of mass velocity (Ẋ, Ẏ, angle ϕ and angular velocity ϕ. (3)
Matlab Based Interactive Simulation Program 3 Up to date segment s ends components are calculated using simple trigonometric equations: x = X + l sinϕ x = X l sinϕ y = Y l cos ϕ y = Y + l cos ϕ (4) (5) Differentiation allows to calculate vertical and horizontal components of segment s ends velocity: ẋ = ẋ = cos ϕ Ẋ + ϕl cos ϕ Ẋ ϕl ẏ = ẏ = sin ϕ Ẏ + ϕl sin ϕ Ẏ ϕl (6) (7) Successive differentiation leads to formulae describing segment s ends acceleration: ẍ = ẍ = cos ϕ Ẍ + ϕl cos ϕ Ẍ ϕl ϕ l sin ϕ + ϕ l sin ϕ ÿ = ÿ = sin ϕ Ÿ + ϕl sin ϕ Ÿ ϕl + ϕ l cos ϕ ϕ l cos ϕ (8) (9) Equations describing state of the segment allow to formulate general relationship between accelerations of the segment s ends and external influences forces and a control torque M: ẍ ÿ ẍ = M coef ÿ F x F y F x F y M (0) Symbol M coef represents coefficients matrix determined by the formulas (8), (9) allowing for equations (), (), (3). 3 Model of Couple of Segments Movement The case of joined segments requires doubling of the state variables set applied for the case of a single segment. Decomposition of forces and torques is shown on Fig. (index U denotes the upper segment whereas L the lower one).
4 H. Josiński et al. Fig.. Decomposition of forces for a couple of segments For the point of osculation of both segments (a joint) the following dependencies are fulfilled: Fx U = Fx L = Fx UL () Fy U = Fy L = Fy UL () M U = M L = M UL (3) where symbols Fx UL, Fy UL were introduced as notation for reaction forces. In the setup of two segments it is very important to prevent any break-up of the segments. For joint ends of both segments ( ) ( ) x U,y U, x L,y L the following conditions should hold: x U (t + t) = x L (t + t) y U (t + t) = y L (t + t) (4)
Matlab Based Interactive Simulation Program 5 Expansion of the functions x U (t), y U (t), x L (t), y L (t) into Taylor series leads to following dependencies (with sufficiently small t): F UL Mcoef L x U (t + t) = x U (t) + ẋ U (t) t + ẍ U (t) t x L (t + t) = x L (t) + ẋ L (t) t + ẍ L (t) t y U (t + t) = y U (t) + ẏ U (t) t + ÿ U (t) t y L (t + t) = y L (t) + ẏ L (t) t + ÿ L (t) t Conditions (4) enable to calculate unknown values of reaction forces F UL (5) (6) (7) (8) y. With this end in view appropriate parts of the coefficient matrices Mcoef U, should be substituted to the general formula (0) bearing in mind the dependencies (), (), (3). Next stage of the extension of multisegment system is setting of couples of segments into a fragmentary and simplified silhouette of a human form named the biped (Fig. 3). x,.5 0.5 0 0. 0. 0 0. 0. 0.3 Fig. 3. Biped example of the multisegment mechanical system In the biped system there are following reaction forces:. From interaction upper leg lower leg (separately for left and right leg).. From interaction upper right leg upper left leg in the joint root. 3. Between the ground and a leg (separately for left and right leg). The biped s digital model was created by means of the Matlab package and used in simulation experiments. The goal of the discussed experiment was to simulate a biped s jump down on the ground. The following values of the model parameters were applied: mass of each segment m =, length of each segment l = 0.5, vertical coordinate of the root yl U (0) = yu R (0) =, angles of vertical inclination of individual segments: ϕ U L (0) = π/5, ϕl L (0) = π π/5, ϕu R (0) = π/6, ϕl R (0) = π π/6.
6 H. Josiński et al. 4 Test Run of a Single Experiment Experiment begins with setting of state variables initial values. Variables new values are calculated in consecutive moments spaced by the actual value of the integration step. This digital model applied ode45 integration method which is one of many Matlab methods for solving ordinary differential equations. It uses 4th and 5th order Runge-Kutta formulas and is based on the Dormand-Prince method [4]. Single integration step determines values of the right sides of the state equations. The calculations are carried out in following stages:. Kinetics of individual ends of the segments calculation of location and velocity.. Dynamics of individual ends of the segments calculation of coefficients matrices. 3. Forces of reaction with the ground at the current stage of the research they are set to 0. 4. Reaction forces in the joints joining segments usage of the Gauss elimination method with the application of coefficients matrices calculated in the point. 5. Dynamics of the centres of the mass of individual segments calculation of accelerations. 6. Drawing of the actual location of the segments. 7. Placing of calculated values of the right sides of the state equations (velocities as values of the appropriate state variables calculated in the previous integration step and accelerations calculated in the point 5) to the appropriate state variables derivatives. 8. Integration by means of the method ode45 gives new values to the state variables. Attainment of the limiting value of simulation time ends the experiment. The aim of this investigation was to find a graph depicting distance between joint ends of adjacent segments. Such distance was calculated as Euclidean distance between two points. For a couple of segments from the Fig. the applied formula is as follows: (x d = U ) ( ) xl + y U y L (9) Graphs acquired by using this formula for left and right knee joints are shown on Fig. 4A and 4B, respectively. For the joint root the formula (9) was modified as follows: d = (x U L xu R) + ( y U L y U R) (0) (indexes L and R denote upper ends of the upper segments of the left leg and the right one, respectively). Graph acquired for the root is shown on Fig 5. The order of magnitude of the segments ends distance read out from the graphs documents the empirical observation that the adjacent segments don t break up.
Matlab Based Interactive Simulation Program 7 A 7 x 0 6 B 6 x 0 6 6 5 5 4 3 4 3 0 0 50 00 50 00 50 300 350 400 450 0 0 50 00 50 00 50 300 350 400 450 Fig. 4. The distance between common segments ends: A) of the left knee B) of the right knee.4 x 0 5. 0.8 0.6 0.4 0. 0 0 50 00 50 00 50 300 350 400 450 Fig. 5. The distance between common segments ends of the root 5 Conclusion This paper aimed to present the method of construction of a simulation program for D multisegment mechanical systems. A model of such a system was created and implemented using Matlab package. Further investigations will include extensions of the model first of all worth mentioning is the problem of biped s various types of gait. Next plans comprise transfer of the model to 3D space and its numerical stability analysis. 6 Acknowledgements This paper has been supported by the project System with a library of modules for advanced analysis and an interactive synthesis of human motion co-financed by the European Regional Development Fund under the Innovative Economy Operational Programme Priority Axis. Research and development of modern technologies, measure.3. Development projects.
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