Introduction to Computational Robotics with SOFA-2009 model

1. Student research project Phoenix-3 Introduction to Computational Roboticswith SOFA-2009 model Alex Astapkovitch, Head of the Student Design Center State University of Aerospace Instrumentation Saint-Petersburg,Russia 2010

2. Computatonal robotics - what is it ? - We have computational chemistry, computational plasma physics and so on So, why we have not a “ Computational Robotics “ ? • Computational robotics is the branch of computer science • related with the control theory ; • From theory point of view computational robotics has as the goal the understanding of investigated case with: • - the robot mode; • - the environment model; • - numerical simulation ; • From practice point of view computational robotics is a efficient tool for the robot control system designer;

3. What for this report is ? -Goal of this report is to illustrate the computational robotics approach; Presented research was focused on neuron net control system: - Virtual model SOFA-2009 was used with two channel neuron net control system with the supervised learning approach; • - Discovered phenomena - learning asymmetry effect ; • - Developed algorithm – modified one step learning • procedure with Lagrange multiplyers method for • linear relation between variables;

4. Cinematics and dynamics models of virtual robot SOFA direction “forward” axe y φ(t) – robot angle position axe x R(t) – instant rotation radius Rс (t)– robot center position vector R0 (t) - instant center of arc axe x earth fixed frame Virtual robot SOFA-2009 -It is as simple as possible model of two wheel robot ; φ (t+∆t)- φ (t) = ∆ φ (for left wheel) = ∆ φ (for right wheel) R0 (t) = R0 (t+∆t) Basic relations:

5. Virtual robot SOFA-2009 - Model includes dynamic equations, gear model for every wheel, motor model, control system model. Model SOFA-2009 is defined with parameters set : Dw = 0.3 Lr = 0.5 Jr = 0.25 k11 = k22 = 75 k12 = k21 = 10 Rm = 0.1 Lm = 0.01 k13 = k23 = 1.5 Vmax = 12 Vmax is the maximal absolute value for accumulator voltage. - The simplest as possible model consists of 7 ODE with at least 9 parameters.

6. Model of the neuron net control system: Sensor layer Left motor control channel WLeft = [w1L…………..w9L] S1 Uin left S2 S3 Uin right Unlimited case S9 Right motor control channel WRight = [w1R………W9R] Vout Sval Unlimited case Smax Vmax = 12 V Vmax Smax Vin Sraw Uin = S*W Actor neuron transfer function Sensor neuron transfer function Actor layer neuron Vmax, Vmin Uout Left Motor Uout Right Motor Virtual robot SOFA-2009

7. Virtual robot SOFA-2009 NAME SENSOR DESCRIPTION FORMULA DF Distance to final point robot center position from instant position D(F)-D(Rc(t)) DV (W1(t)+ W2(t))/2 Instant linear velocity Difference between the robot inclination angle at the final point and the instant one φ (F)- φ (t) AF AV Instant robot axe inclination angle rotation speed Dw/2*Lr*( W2(t)- W1(t)) Difference between the robot angle speed at the final point and the instant one d φ - Dw/2*Lr*( W2(t)- W1(t)) dt ( F) AVF W1(t),W2(t) W1(t),W2(t) Instant rotation speed for left and right wheels Difference between rotation speed at the final point for left (right) wheel and the instant one W1(F)- W1(t) W2(F)- W2(t) W1F,W2F Note: F denotes the final point

8. S1(T1) S2(T1) .. Sn(T1) S1(T2) S2(T2) .. Sn(T2) …………………… S1(Tp) S2(Tp) .. Sn(Tp) w1 w2 wn Ua1(T1) Ua1(T2) Ua1(Tp) = * Supervized learning- “Learning” stands for the procedure of determination of weights; One step supervised learning (simplified form) S*w = Ua - the bad posed problem for w S,U – learning sample set, w – unknown vector Tichonov regularization formulation provides stable solution min F(w) = (Sw - Ua, Sw – Ua) +  (w,w) w Weights calculation with one step learning procedure : w = (ST S +  E) –1 ST Ua

9. 1. SAMPLE GENERATING AND NEURON NET LEARNING Robot model Cauchy problem solution for [T0 -T1] Initial position vector X0 Control voltage matrix (vector Ua(t) for every motor ), that corresponds to robot mission SOLUTION TABLE [ti, X (ti) ] NEURON NET CONTROL SYSTEM STRUCTURE Final position vector X(T1), velocity vector V(T1) Sensor System Model Weight Matrix Calculation W= (StS+γE) -1St Ua (one step procedure) Supervised learning procedure • Experiment with virtual robot includes at least three steps: • sample set generating and neuron net control system learning ; • simulation of the robot dynamics with "learned "neuron net control system; • research experiments;

10. Supervised learning procedure 2. CONTROL SIMULATION Cauchy problem solution for autonomous ODE problem Weights, received from learning procedure Initial position Robot model cinematic and dynamic model NEURON NET CONTROL SYSTEM MODEL Ua = s(t)*w Final position S(t) - Sensor model 3. NUMERICAL EXPERIMENTS Cauchy Problem Solution Initial and final positions, control net structure depends on research PROBLEM POST PROCCESINGS

11. Virtual robot SOFA-2009 Sample of autonomous operation for π rotate task: rotation to left on π with limited and unlimited Vmax motor currents for limited and unlimited voltage phase portrait for unlimited case: start point (0,0),final (3.14,0) phase portrait for limited Vmax: start point (0,0),final (3.14,0)

12. Y X Learning sample set : “Robot has to reach the prescribed point and stop at it” Learning asymmetry effect SOFA neuron net control system was learned with different samples: Simple behavior 1 - Rotation in place on π/ 4 to left 2 - Moving from the point (0.0) to the point(- 4,4) Complex behavior 3 - Rotation in place on π/ 4 to left and moving from point (0.0) to the point (- 4,4) 4 - Rotation in place on π/ 4 to left and moving from point (0.0) to the point (- 4,4) and rotation in place on π/ 4 to right and moving from point (0.0) to the point (4,4) Generalized learning procedure 5 - Rotation in place on π/ 4 to left and moving from point (0.0) to the point (- 4,4). Robot was learned with using modified learning procedure with Lagrange multipliers to provide the symmetry for weights absolute meanings

13. Learning asymmetry effect - WEIGHTS REFLECT ASYMMETRY OF USED LEARNING SAMPLE SET 3 - Rotation in place on π/ 4 to left and moving from point (0.0) to the point (- 4,4) 4 - Rotation in place on π/ 4 to left and moving from point (0.0) to the point (- 4,4) and rotation in place on π/ 4 to right and moving from point (0.0) to the point (4,4) 5 - Rotation in place on π/ 4 to left and moving from point (0.0) to the point (- 4,4). Robot was learned with using modified learning procedure with Lagrange multipliers to provide the symmetry for weights absolute meanings

14. Learning asymmetry effect • From comparison of the 3 an 4 sample it can be concluded, that • the learning asymmetry effect exists; • Effect results in the asymmetry of the weight values of the • symmetry control channels if asymmetry sample is used for • learning.To avoid this effect the symmetrical learning • set has to be used; • For real robot the situation is more complex. It is clear, that the • experimental data will put on weight the experimental asymmetry • effect ; • It means, that the learning asymmetry effect and the experimental • asymmetry effect influences have to be separated from the real • asymmetry of control channels;

15. Modified One Step Learning Procedure • The one possible way to solve asymmetry problem is using the description of the relation between the weights in explicit form; • For learning asymmetry there are exist linear relation between the same weights in the different channels, so it is possible to use Lagrange multiplier method; • One step learning procedure on base of Lagrange multipliers method is proposed ( learning sample 5), that provide possibility to take into account the existence of the linear relations for weights and avoid asymmetry effects also;

16. Modified One Step Learning Procedure Let us Wk is the weight vector for k-th control channelWk = [ wk1, wk2, ……. wk Nsen ]T In this case the weight vector for whole system can be expressed as W = [ W1 W2 Wk Wk WNc ]’ Actor vector can be expressed with the same manner Uak = [ Uak1 Uak2 ……. Uak Np ]’ Ua = [ Ua1 Ua2 ..... UaNс ]’ Learning asymmetry problem can be solved if one take into account the two type of linear relations between weightswkj = wmj or wkj = - wmj In common way the set of this relation can be presented as L W = b With the introduced above vectors Lagrange multipliers method for one step learning procedure can be formulated as linear programming optimization problem: min F(W) = (SW - Ua, SW – Ua) +  (W,W) + Dμ LW W,Dμ

17. Modified One Step Learning Procedure The elegant form of the one step learning procedure exists, if the modified vectors W and Ua are used; Let us μ is the vector that is formed from Lagrange multiplier μ = [μ1, μ2, ……. μ Nsen ]’ and let us introduce the modified vector of the independent variables Wμ = [ W1 W2 Wk Wk WNc μ]’ Vector Ua has to be modified with the same manner and the resulted vector will denote as Uμ. This vector is formed from vector Ua and the added vector b. With this vectors the one step learning procedure can be expressed as -1 Wμ= (Sμ’Sμ+ Eμ) Sμ’Uμ • This method was tested for the effect of asymmetry learning problem, described above. The results of the learning with this procedure are presented in the table (learning sample 5). So, case 4 and case 5 supports each other !

18. Supporting publications • Astapkovitch A.M. Learning Asymmetry Effect for the Neuron Net Control Systems (to be published) 2. Astapkovitch A.M. Virtual mobile robot SOFA-2009 Proc. International forum “Information and communication technologies and higher education - prioriries of modern society development”, p.7-15,SUAI Saint-Petersburg, 2009 3. Astapkovitch A.M. Оne step learning procedure for neural net control system. Proc. International forum “Information systems. Problems, perspectives , innovation approaches” , p.3-9,SUAI Saint-Petersburg, 2007 • MathCAD and MathLab examples library “Virtual robot SOFA-2009 with neural net control system” can be downloaded for free from the site http://guap.ru/guap > student design center > student projects > SOFA-2009