Optimization and Synthesis of a Robot Fish Motion

Inverse method algorithm for investigation of mechatronic systems in vibration technology is used for robotic systems motion control synthesis. The main difference of this method in comparison with simple analysis method is that before synthesis of a real system the optimal control task for abstract initial subsystem is solved [1 4]. As a result of calculations the optimal control law is found that allows to synthesize series of structural schemes for real systems based on initial subsystem. Is shown that near the optimal control excitation new structural schemes may be found in the medium of three kinds of strongly non–linear systems: systems with excitation as a time function; systems with excitation as a function of phase coordinates only; systems with both excitations mixed [2 – 4]. Two types of vibration devices are considered. The first one is a vibration translation machine with constant liquid or air flow excitation. The main idea is to find the optimal control law for variation of additional surface area of machine working head interacting with water or air medium. The criterion of optimization is the time required to move working head of the machine from initial position to end position. The second object of the theoretical study is a fin type propulsive device of robotic fish moving inside water. In that case the aim is to find optimal control law for variation of additional area of vibrating tail like horizontal pendulum which ensures maximal positive impulse of motion forces acting on the tail. Both problems have been solved by using the Pontryagin’s maximum principle. It is shown that the optimal control action corresponds to the case of boundary values of area. One real prototype was investigated in linear water tank.


Introduction
Inverse method algorithm for investigation of mechatronic systems in vibration technology is used for robotic systems motion control synthesis. The main difference of this method in comparison with simple analysis method is that before synthesis of a real system the optimal control task for abstract initial subsystem is solved [1 -4]. As a result of calculations the optimal control law is found that allows to synthesize series of structural schemes for real systems based on initial subsystem. Is shown that near the optimal control excitation new structural schemes may be found in the medium of three kinds of strongly non-linear systems: -systems with excitation as a time function; -systems with excitation as a function of phase coordinates only; -systems with both excitations mixed [2 -4]. Two types of vibration devices are considered. The first one is a vibration translation machine with constant liquid or air flow excitation. The main idea is to find the optimal control law for variation of additional surface area of machine working head interacting with water or air medium. The criterion of optimization is the time required to move working head of the machine from initial position to end position. The second object of the theoretical study is a fin type propulsive device of robotic fish moving inside water. In that case the aim is to find optimal control law for variation of additional area of vibrating tail like horizontal pendulum which ensures maximal positive impulse of motion forces acting on the tail. Both problems have been solved by using the Pontryagin's maximum principle. It is shown that the optimal control action corresponds to the case of boundary values of area. One real prototype was investigated in linear water tank.

Translation motion system
First object with one degree of freedom x and constant water or air flow V 0 excitation is investigated (Fig. 1., 2.). The system consists of a mass m with spring c and damper b. The main idea is to find the optimal control law for variation of additional area S(t) of vibrating mass m within limits (1.):

SS tS
www.intechopen.com The criterion of optimization is the time T required to move object from initial position to end position. Then the differential equation for large water velocity Vx ≥ 0  is (2): where ut St k =⋅ () () , c -stiffness of a spring, b -damping coefficient, V 0 -constant velocity of water, S(t) -area variation, ut () -control action, k -constant. It is required to determine the control action u = u(t) for displacement of a system (2) from initial position x(t 0 ) to end position x(t 1 ) in minimal time (criterion K) K= T, if area S(t) has the limit (1).  To assume tt T == 00 ; 1 , we have KT = .
From the system (2), we transform xx xx == 11 2 ;  or and we have Hamiltonian (3): Scalar product of those vector functions ψ and X in any tim e (fu nction H) m u st be maximal. To take this maximum, control action u(t) must be within limits ut u ut u ==

Theoretical synthesis of control action
For realizing of optimal control actions (in general case) system of one degree of freedom needs a feedback system with two adapters: one to measure a displacement and another one to measure a velocity. There is a simple case of control existing with only one adapter when directions of motion are changed (Fig. 5) [3]. It means that control action is like negative dry friction and switch points are along zero velocity line. In this case equation of motion is (6): where m -mass; c, b, k, V0 -constants. Examples of modeling are shown in Fig. 6. -Fig. 9.
x2 S1 x(0) x1 x(T) S1 S2 S2 x1 S1 S1 S2 x(0)  A search for the case with more than one limit cycle was investigated in a very complicated system with linear and cubic restoring force, linear and cubic resistance force and dry friction. It was found that for a system with non-periodic excitation (like constant velocity of water or air flow) more than one limit cycles exist ( Fig. 10., 11.).
where A2, a -constants, v, x -velocity and coordinate of moving mass m2.  It is shown that adaptive systems are also very stable because of water or air excitation and damping forces depending from velocity squared. In order to get values of parameters in equations, experiments were made in the wind tunnel. It should be noted that the air flow is similar to the water flow, with one main difference in the density. It is therefore convenient to perform synthesis algorithm experiments done with air flow in the wind tunnel. Experimental results obtained in the wind tunnel to some extent may be used to analyze water flow excitation in robot fish prototypes. Using "Armfield" subsonic wind tunnel several additional results were obtained. For example, drag forces, acting on different shapes of bodies, were measured and drag coefficients were calculated by formula dd V FCS where ρ -density, V -flow velocity, S -specific area, C d -drag coefficient. These coefficients depend on bodie's geometry, orientation relative to flow, and non-dimensional Reynolds number.

Investigation of a rotating system
The second object of the theoretical study is a fin type propulsive device of robotic fish moving inside water. The aim of the study is to find out optimal control law for variation of additional area of vibrating tail like horizontal pendulum, which ensures maximal positive impulse of motion forces components acting on a tail parallel to x axis (Fig.18). Problem has been solved by using Pontryagin's maximum principle. It is shown that optimal control action corresponds to the case of boundary values of area. Mathematical model of a system consists of rigid straight flat tail moving around pivot (Fig.  18.,19.). Area of the tail may be changed by control actions. For excitation of the motion a moment M(t,φ,ω) around pivot must be added, where ω = dφ/dt -angular velocity of a rigid tail. To improve performance of a system additional rotational spring with stiffness c may be added (Fig. 19.).  The principal task described in this report is to find optimal control law for variation of additional area B(t) of vibrating tail within limits (7.): After integration from equations (9) and (10) we have (11,12): Then the criterion of optimization (full impulse) is: Equations (13) will be used to solve the optimization problem.

Task of optimization
Solution of optimal control problem for a system with one degree of freedom (11) by using the Pontryagin's maximum principle includes following steps [8 -13]: 1. Formulation of a criterion of optimization (13): 2. Transformation of the equation (9) and equation (13) in the three first order equations with new variables φ0, φ1 and φ2 (phase coordinates): ΦΦΦ 012 ,, -right side of system equations (14).
For functions ΦΦΦ 012 ,, calculations are following differential equations [9 -12]: where left side is the derivation in time t: d dt From equations (15) and (18) we get nonlinear system of differential equations to find functionsψψψ 012 ,, . Solution of such system is out of the scope of this report because it depends from unknown moment M(t,φ,ω). But some conclusions and recommendations may be given from Hamiltonian if excitation moment M(t,φ,ω) does not depend from phase coordinates φ = φ1, ω = φ2: In this case scalar product of two last vector functions ψ and Φ in any time (Hamiltonian H [11]) must be maximal (supremum -in this linear B case) [8 -12]. To have such maximum (supremum), control action B(t) must be within limits Bt B Bt B == 12 () ; () , depending only from the sign of a function (19) or (20)  ( 2 ) 4 (20)

B = B1;
From inequalities (19) and (20) in real system synthesis following quasi-optimal control action may be recommended (21)

Synthesis of mixed system with time-harmonic excitation and area adaptive control
In the case of time-harmonic excitation moment M in time domain is (see equations (9) and (11) (9)). Practically angular acceleration of the tail reaches steady-state cycle after one oscillation  (12)). Impulse is non-symmetric against zero level (non-symmetry is negative)  Force of pivot to push a hull (necessary for robotic fish motion)

Synthesis of a system with adaptive excitation and adaptive area control
In a case of adaptive excitation a moment M may be used as the function of angular velocity in the form [3. 4]: Results of modeling are shown in Fig. 26.    (9)). Typically angular acceleration of the tail reaches steady-state cycle after half oscillation   (12)). Impulse is non-symmetric against zero level (non-symmetry is negative) Fig. 30. Impulse Ax(t) as a function of angle φ Impulse is non-symmetric against zero level (non-symmetry is negative), it means that criterion of optimization (8) is positive and force of pivot pushes fish hull to the right.

Robot fish model
A prototype of robot fish for experimental investigations was made (Fig. 32, 33). This prototype was investigated in a linear water tank with different aims, for example: -find maximal motion velocity depending on the power pack capacity; -find minimal propulsion force, depending on the system parameters.
Additionally this prototype was investigated in a large storage lake with autonomy power pack and distance control system, moving in real under water conditions with waves ( Fig.  34.). The results of the theoretical and experimental investigation may be used for inventions of new robotic systems. The new ideas of synthesising robotic systems in Latvia can be found in [15 -23].

Conclusion
Motion of robotic systems vibration by simplified interaction with water or air flow can be described by rather simple equations for motion analysis. That allows to solve mathematical problem of area control optimization and to give information for new systems synthesis. Control (or change) of object area under water or in air allows to create very efficient mechatronic systems. For realization of such systems adapters and controllers must be used. For this reason very simple control action have solutions with use of sign functions. Examples of synthesis of real mechatronic systems are given. As one example of synthesis is a system with time-harmonic moment excitation of the tail in the pivot. The second example of synthesis is a system with adaptive force moment excitation as the function of phase coordinates. In both systems area change (from maximal to minimal values) has control action as the function of phase coordinates. It is shown that real controlled systems vibration motion is very stable.