Control Optimization of a Robotic Bird

CONTROL OPTIMIZATION OF A ROBOTIC BIRD Micael S. Couceiro* N. M. Fonseca Ferreira*, J. A. Tenreiro Machado** *Institute of Engineering of Coimbra Portugal, (e-mail: [email protected], [email protected]) ** Institute of Engineering of Porto Portugal, (e-mail: [email protected]) Abstract: In this paper it is studied the modeling and control of a robotic bird. The results are positive for the design and construction of flying robots and a new generation of airplanes developed to the similarity of flying animals. The development of computational simulation based on the dynamic of the robotic bird should allow testing strategies and different algorithms of control such as integer and fractional controllers. Firstly, we provide an overview of the system model; secondly, we compare the behaviour of fractional and integer order controllers using different algorithm for the controller optimization in order to obtain the minimum error. Keywords: robotic, bird, fractional controller, dynamic, position control, flapping flight, PSO. 1. INTRODUCTION This paper presents the development of a dynamical model of a bird with several control techniques. We compare the results of integer and fractional order control algorithms presenting some real-time experiments of the closed loop system. Birds have many similar characteristics to the reptiles but they are different from all the other animals because of their feathers (Cianchi, et al. 1988) and other unique characteristics studied (Kenneth, Randall, and Terry Dial et al. 2006). In order to simulate and implement a robotic bird we would need to consider every single physical aspect. Having as inspiration the behaviors of the animals, some works have been developed with the purpose of implementing similar robotic behaviors. Examples of some robot-animals already build are spiders (Vallidis et al. 2001) and snakes (Spranklin et al. 2006). Both of them require an extended study of the physics and behavior of the real animals. Other interesting works focus specific characteristics of animals applying new technologies such as morphing materials in order to create wings (Manzo et al. 2006). The paper is organized as follows. Section two, presents the state of the art. Section three describes the bird dynamics implemented in our model. In the section four we will present the architecture of the robot control and section five will give an approach to the optimization methods used to tune the controllers. In section six we will compare the performance of the different controllers. Finally, outlines the main conclusions in section seven.

creatures. The first ideas to implement biological inspired flying vehicles date from the XV century. Leonardo da Vinci began to work on a machine, powered by muscular activity, which would allow a man to hover in the air by moving its wings like birds do. There are drawings which show the various kinds of “ornitotteri”, the flying machines designed by Leonardo (Fig. 1). The “ornitotteri” was a significant outgrowth of Leonardo’s studies on the anatomy of bird’s wings and on the analysis of the function and distribution of its feathers (Cianchi et al. 1988), but it took until 1870 for the first successful ornithopter to be flown 70 meters by its builder Gustave Trouvé (Chronister et al. 2007). Our work implements a system that includes a physical and dynamic model of a bird (Zhu, Muraoka, Kawabata, Cao, Fujimoto and Chiba et al. 2006) that use a set of equations to simulate the behavior of a bird, using a real time animated model taking aerodynamics into consideration. In this model, a bird flies by the wing beat motion, using its tail feathers. Besides, the trajectory is established by determined points in the space adjusting the bird’s orientation and flapping such that the bird passes through these points in sequence. This allows the bird to fly along an arbitrary path.

2. STATE OF THE ART The first concepts in the area of biological inspired flying locomotion are quite old. Humans have always been fascinated with birds and other flapping wing

Fig. 1. Drawings of Leonardo da Vinci studies for an human powered “ornitotteri”

A method of producing realistic animations (Fig. 2) from numerical solutions is given for generic bird models with various levels of complexity (Parslew et al. 2005). The study describes the development of models, implemented in the analysis of flapping flight, balancing the scientific analysis and model-based animation. The presented results show numerical data and visual simulations able to produce realistic flapping flight with physical strong foundations. The aerodynamic coefficients of lift and drag are based on the Blade-Element Theory. This method requires an input of the angle of attack, along with the key aerodynamic properties of the wing in order to determine the lift and drag coefficient.

A bird's wing is curved along the top, so that when air passes over the wing and divides, the curve forces the air on top to travel a greater distance than the air on the bottom (Fig. 4). The tendency of airflow is to correct for the presence of solid objects and to return to its original pattern as quickly as possible. Hence, when the air hits the front of the wing, the rate of flow at the top increases to compensate for the greater distance it has to travel than the air below the wing. And as shown by Bernoulli’s Principle, fast-moving fluid exerts less pressure than slow-moving fluid; therefore, there is a difference in pressure between the air below and the air above. Whenever such a pressure difference exists in nature, a force is created in the direction of the lower pressure, in other words, downward. Newton’s Third Law of Motion states that for every action there is an equal and opposite reaction. This is another way of saying you can’t exert a force without something to push against. So, if a wing deflects air downward there must exist an equal and opposite reaction above and below the wing keeping it aloft. Force Lower Pressure

Fig. 2. Animations based on numerical solutions. Birds’ tails also play an important aerodynamic role in mechanical flight power and flight performance (Evans, Rosén, Park and Hedenström et al. 2001). Theory provides a conventional explanation for how bird’s tail works. In (Zhu, Muraoka, Kawabata, Cao, Fujimoto, and Chiba et al. 2006) the influence of the tail and feathers is taken in consideration (Fig. 3).

Fig. 3. Bird following a trajectory. 3. SYSTEM MODEL In order to establish a methodology and appropriated strategy to this project, it is important to study the aerodynamic principles knowing that those will be crucial to the physics behind the flight of birds. 3.1. Aerodynamics All the aerodynamics behind wings could be a study to approach in an entire paper or more. In this section we will give a basic explanation of how wings make it possible for the bird to fly.

Faster Airflow Slower Airflow

Higher Pressure Fig. 4. Aerodynamics of wings. With some research, we can find some applications which allow us to study the aerodynamic characteristics of some typical airfoil profile. One of those free applications is named FoilSim which is a educational package created by NASA to help and instruct students with the principles of aerodynamics. FoilSim can show us charts with the forces and aerodynamic velocity acting in the airfoil. Although the majority of avian flight studies have focused on the wings, the tail also appears to be crucial to the evolutionary success of birds as flying organisms. The precise use of the tail in flying birds has not been thoroughly documented (Zhu, Muraoka, Kawabata, Cao, Fujimoto and Chiba et al. 2006). The tail feathers are instrumental in stabilizing the flight, changing the direction of the forward movement, compensating for the lift force, and acting as a brake when the bird lands. We are using the tail in order to cause a drag force changing the moment of the bird and, consequently, producing a rotation around an axis equal to the axis of rotation of the tail. That is, if the tail is bending up, the bird will rotate around the same joint bending up too. If the tail bends up and twists right for example (Fig. 5), the bird will then rotate around both the joints of the

tail up and right. The angle of rotation of the tail is always relative to the movement of the bird.

coefficients as functions of the angle of attack of the local wind (Fig. 7). (4) (5)

y - axis

The wing aerodynamics properties of maximum lift C l m ax and drag C d m ax coefficients and zero drag C d 0 coefficient since we are not considering any particular wing aerodynamics at this point.

z- axis Fig. 5. Action of bird tail.

(6) (7) (8)

The influence of the tail has relevance in the calculation of the moments (Berg and Rayner et al. 1995). The bird will be able to rotate using the tail or using different angles of attack on each wing (or different velocities on each wing). 3.2. Gliding Flight The relative wind acting on a wing produces a certain amount of force which is called the total aerodynamic force. This force can be resolved into components, called Lift and Drag (Fig. 6).

Cd(α)

Cl(α)

L - Lift. D - Drag. V - Velocity of air flow relative to wing.

Fig. 6. Force acting on the wing.

Fig. 7. Lift and Drag coefficients functions.

The Lift L (2) is the component of aerodynamic force perpendicular to the relative wind and the Drag D (3) is the component of aerodynamic force parallel to the relative wind. Those components can be expressed by the following formulae:

Some birds take advantage of the air currents to remain aloft for long periods without flapping their wings (Fig. 8). Gliding has a lower metabolic cost than flapping flight (Baudinette, Schmidt-Nielsen et al. 1974). The bird’s aerodynamic characteristics determine how far and for how long it can glide, and how successfully it can soar in moving air.

(2) (3) The Lift and Drag on the wing depends on the wing area S, the density of air ρ, the velocity of the air flow relative to the wing v∞ and the Lift and Drag coefficients Cl and Cd respectively, expressed as functions of the angle of attack α. The Lift and Drag coefficients depend on the shape of the airfoil and will alter with changes in the angle of attack and other wing trimmings. The characteristics of any particular airfoil section can conveniently be represented by graphs showing the amount of lift and drag obtained at various angles of attack, the lift-drag ratio, and the movement of the center of pressure. Similarly to Parslew et al. 2005 we adopted the bladeelement theory representing the Lift (4) and Drag (5)

Fig. 8. Gliding flight of birds. Those aerodynamic characteristics can be optimized by the bird in the flight by changing the wing spans and wing areas.

For optimal gliding a bird’s wing must maximize lift and minimize drag. As a rule, the smaller the bird, the shorter the distance it can glide and the faster it sinks. During gliding the wings are stretched out stiffly. A good glider travels a long way horizontally with minimum loss of height, but eventually loses altitude due to the pull of gravity. The efficiency of a glider can be measured by calculating the angle between the track of its motion and the horizon. This angle depends not upon the weight of the bird but rather upon the forces of lift and drag, through wing shape does have some influence. 3.3. Flapping Flight The aerodynamics involving flapping wings differs in many ways from conventional aerodynamics, but some conventional rules apply. A conventional airplane uses a propeller for thrust and fixed wings for lift. An ornithopter’s wing must provide both of these forces.

(6)

(5)

During flapping flight, flying animals invest power to move the air (aerodynamic power) and to move the wings (inertial power). The interaction between a flapping wing and the air is very complicated since a bird must flap its wings to generate lift and thrust to overcome gravity and drag. The forces generated by this interaction are chaotic and their simulation is often unstable because of high sensitivity. The forces on the wing (Berg and Rayner et al. 1995) vary throughout the flapping cycle as we will see in the dynamical analysis (Fig. 9). On the downstroke air is displaced in a downward and backward direction. On the upstroke, the situation is reversed being the area of the wing smaller than before in order to make a positive global thrust force. In order to make the area of the wing smaller, birds uses different techniques such as manipulating the wings. To simplify we considered the area in the upstroke half the area in the down stroke.

(3)

(4)

(2)

(1)

Fig. 9. Flapping flight sequence. Downstroke: 1-2-3; Upstroke: 4-5-6. As seen previously, the forces of Lift and Drag will depend on the angle of attack. However, which will be the behavior of these forces when flapping wings? As Parslew et al. 2005 we considered the existence of an advance angle related with the flapping velocity and the freestream velocity (9). (9) The advance angle will then be zero when the velocity of the wings is zero falling in the situation analyzed previously in gliding flight. In other words, this means that through the angle of attack it is possible to control the amplitude of the forces of Lift and Drag. On the other hand, the angle that these forces have relatively to the air flow can be controlled through the flapping velocity. There are two possible states while flapping the wings: The downstroke and upstroke as we can see in Fig. 10.

a)

Fig. 10. a) Downstroke, b) Upstroke

b)

If the wing is placed into a flow velocity, v∞, a thrust force will develop due to the horizontal component of the Lift that appears in the downstroke. So, in order to have a positive thrust, the wing will have to increase its velocity to overcome the opposing horizontal force originated in the aerodynamic Drag. A greater flapping speed velocity corresponds to a greater advance angle originating a greater thrust force. The horizontal (x-axis) and vertical (z-axis) forces are related with the Lift, Drag and advance angle by the following equations: (10) 4. BIRD MIMO CONTROLLERS Conventional controllers such PID and many advanced control methods are useful to control linear processes. In practice, most processes are nonlinear. If a process is a just little nonlinear, it can be treated as a linear process. If a process is severely nonlinear, it can be extremely difficult to control. Nonlinear control is one of the biggest challenges in modern control theory. While linear control system theory has been well developed, it is the nonlinear control problems that present the most headaches. Nonlinear processes are difficult to control because there can be so many variations of the nonlinear behavior.

The first attempt to control our system will be changing the wing speed velocity, angle of attack and tail

rotations accordingly to the position error (Fig. 11).

Fig. 11. Control Diagram. 4.1. Integer PID Controller The PID combines the advantages of PI and PD controller. The integral action is related with the precision of the system being responsible for the permanent regime error. The unstable effect of PI controller is counterbalanced by the derivative action that tends to increase the relative stability of the system at the same time that makes the system response faster due to its anticipatory effect.

five parameters, and the derivative and integral orders improve de design flexibility. The mathematical definition of a derivative of fractional order has been the subject of several different approaches. For example, we can mention the Laplace and the Grünwald-Letnikov definitions (14) (15a)

The PID action is given by: (15b)

(11) whose Laplace transformed is: (12) where E(s) is the error signal and U(s) is the controller’s output. The parameters K, Ti and Td represent, respectively, the proportional gain, the integral time and the derivative time. The design of the PID controller consist on the determination of the optimum PID parameters (K, Ti, Td) that minimize J, the integral of the square error (ISE), defined as:

where is the gamma function and h is the time increment. Grünwald-Letnikov definition is perhaps the best known one being more suitable for the realization of discrete control. In our case, for implementing FO (Fractional Order) algorithms of the type, (16) we adopt a 4th-order discrete-time Pade approximation (ai, bi, c i, di , k = 4):

(13) where i(t) is the step response of the closed-loop system with the PID controller and ir(t) is the desired step response. 4.2. Fractional PID Controller The fractional order controllers are controllers whose dynamic behavior is described with differential equations whose order is not an integer number. In other words, the fractional order PID controllers’ have

(17) where KP are the position loop gain. Fractional PID’s are also known as PIλDμ controllers. If both λ and μ are 1, the result is a usual PID (henceforth called integer PID as opposed to a fractional PID). If λ = 0 (Ti = 0) we obtain a PDμ controller. All these types of controllers are particular cases of the PIλDμ controller (Fig. 12).

It can be expected that PIλDμ controller may enhance the systems control performance due to more tuning knobs introduced. Actually, in theory, PIλDμ itself is an infinite dimensional linear filter due to the fractional order in differentiator or integrator. For controller tuning techniques, refer to (Monje, Vinagre, Chen, Feliu, Lanusse, and Sabatier et al. 2004; Liang, Chen, and Fullmer et al. 2004).

Gradient descent is based on the observation that if the real-valued function F(x) is defined and differentiable in a neighborhood of a point a, then F(x) decreases fastest if one goes from a in the direction of the negative gradient of F at a. It follows that, if (18) for γ > 0 a small enough number, then F(a)≥F(b). With this observation in mind, one starts with a guess x0 for a local minimum of F, and considers the sequence x0,x1,x2,… such that (19) We have (20)

Fig. 12. PIλDμ controller performance. Actually, in theory, PIλDμ itself is an infinite dimensional linear filter due to the fractional order in the differentiator or integrator. It should be pointed out that a band-limit implementation of FO is important in practice, i.e., the finite dimensional approximation of the FO should be done in a proper range of frequencies of practical interest 5. OPTIMIZATION METHODS The value of PID terms depends on characteristics of the process and must be tuned accordingly to yield satisfactory control. Properly tuned PID controllers provide adequate control for a large portion of applications. By tuning the constants in the PID or the PIλDµ controller algorithm, the controller can provide control action designed for specific process requirements. The response of the controller can be described in terms of the responsiveness of the controller to an error, the degree to which the controller overshoots the setpoint and the degree of system oscillation. Note that the use of the PID algorithm for control does not guarantee optimal control of the system or system stability. This tuning, or optimization, can occur at a number of levels. At the highest level, the design may be optimized to make best use of the available resources. The implementation of this design will benefit from the use of efficient algorithms (Jan A. Snyman et al. 2005) and the implementation of these algorithms will benefit from writing good quality code. 5.1. The Gradient Descent Optimization Methods To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point.

so hopefully the sequence (xn) converges to the desired local minimum. Note that the value of the step size γ is allowed to change at every iteration. This process is illustrated in the Figure 13. Here F is assumed to be defined on the plane, and that its graph has a bowl shape. The blue curves are the contour lines, that is, the regions on which the value of F is constant. A red arrow originating at a point shows the direction of the negative gradient at that point. Note that the (negative) gradient at a point is orthogonal to the contour line going through that point. We see that gradient descent leads us to the bottom of the bowl, that is, to the point where the value of the function F is minimal.

x1

x3 x2

x4

x0 Fig. 13. Gradient Descent Method. 5.2. The Pattern Search – Lewis and Torczon GPS Generalized pattern search (GPS) algorithms (Audet, Charles and Dennis et al. 2003) are derivative free methods for the minimization of smooth functions, possibly with linear inequality constraints. Examples of pattern search algorithms are the coordinate search algorithm, the pattern search algorithm of Hooke and Jeeves, and the multidirectional search algorithm of Dennis and Torczon (Dennis and Torczon et al. 1991). What they all have in common is that they define the construction of a mesh, which is then explored according to some rule, and if no decrease in cost is

obtained on mesh points around the current iterate, then the mesh is refined and the process is repeated. In 1997, Torczon was the first to show that all the existing pattern search algorithms are specific implementations of an abstract pattern search scheme and to establish that for unconstrained problems with smooth cost functions, the gradient of the cost function vanishes at accumulation points of sequences constructed by this scheme. Lewis and Torczon extended her theory to address bound constrained problems and problems with linear inequality constraints. In both cases, convergence to a feasible point x* satisfying (F(x*), x - x*)≥ 0 for all feasible x is proven under the condition that F(x) is once continuously differentiable. As said before, GPS algorithms essentially focused to smooth functions but they can have a good response to nonsmooth problems too. 5.3. The Simplex Method – Nelder-Mead Since its publication in 1965, the Nelder-Mead (NM) simplex algorithm (Nelder and Mead et al. 1965) has become one of the most widely used methods for nonlinear unconstrained optimization. The NM method (Lagarias, J. C., Reeds, M. H. Wright and P. E. Wright et al. 1998) attempts to minimize a scalar-valued nonlinear function of n real variables using only function values, without any derivative information falling in the general class of direct search methods. The method uses the concept of a simplex, which is a polytope of N + 1 vertices in N dimensions; a line segment on a line, a triangle on a plane, a tetrahedron in three-dimensional space and so forth (Fig. 14).

Optimal solution

simulation is expensive, it is important to make good decisions about where to look. NM generates a new test position by extrapolating the behavior of the objective function measured at each test point arranged as a simplex. The algorithm then chooses to replace one of these test points with the new test point and so the algorithm progresses. Like all general purpose multidimensional optimization algorithms, NM occasionally gets stuck in a rut. The standard approach to handle this is to restart the algorithm with a new simplex starting at the current best value. This can be extended in a similar way to simulated annealing to escape small local minima. 5.4. Particle Swarm Optimization Particle swarm optimization (PSO) algorithm was developed by Kennedy and Eberhart et al. 1995. This optimization technique, based on a population search, is inspired by social behavior of bird flocking fish schooling. An analogy is established between a particle and an element of swarm. These particles fly trough the search space following current optimum particles. In each algorithm iteration, a particle movement is characterized by two vectors representing its current position x and velocity v. The velocity, of a particle, is changed according the cognitive knowledge b (the best solution found so far by the particle) and the social knowledge g (the best solution found by the swarm). The weight of each knowledge, in the velocity update, is different according with the random values i, i = {1, 2}. Theses values are random factor follow a probabilistic uniform low i ~U[0, i max]. Initialize Swarm repeat forall particles do Calculate fitness f end forall particles do vt+1 = I vt + 1(b-x) + xt+1 = xt + vt+1 end until stopping criteria

2(g-x)

Fig. 15. Particle Swarm Optimization. Starting vertex Fig. 14. NM Simplex Method. The method approximately finds a locally optimal solution to a problem with N variables when the objective function varies smoothly. For example, a suspension bridge engineer has to choose how thick each strut, cable, and pier must be. Clearly these all link together, but it is not easy to visualize the impact of changing any specific element. The engineer can use the NM method to generate trial designs which are then tested on a large computer model. As each run of the

where I and t are the inertia and the iteration time respectively. PSO is a very attractive technique among many other population based algorithms, because it has only a few parameters to adjust. PSO have been used with success in a vast number of applications, such as robotics (Tang, Zhu and Sun et al. 2005; Pires, Oliveira, Machado and Cunha et al. 2006) and electrical systems (Alrashidi and El-Hawary et al. 2006). In this paper the PSO will be adopted in the controller tuning.

We will now realize some experiments comparing different optimized controllers in order to achieve the better algorithm of control to use in our model. To tune the controllers we first used a medium scale Gradient Descent method with 200 maximum iterations. The first attempt to control our system will be changing the wing speed velocity, angle of attack and tail rotations accordingly with the cartesian position error. In order to study the system dynamics, during the contact we apply, separately, rectangular pulses, at the references. The trajectory used to optimize the controllers is then a straight line flight with a velocity of vx = 3 m/s during the first 20 seconds. The bird will then need to instantaneously achieve a velocity of vx = 5 m/s. Finally, 20 seconds later, he will instantaneously reduce is velocity to vx = 3 m/s again. In this kind of optimization it is unnecessary the use of a controller in the y-axis since there will be no movement in this axis; therefore we will ignore it for now. Let us first compare the PID and PIλDμ controllers with Gradient optimization. Under the last conditions we obtained the PID and PIλDμ controller parameters from Table 1. Table 1. PID and PIλDμ controller parameters with Gradient Optimization PID PIλDμ

KpX 5 13

KiX 0.1 0.1

KdX 7 3

µX 0.9

λX 1

KpZ 100 30

KiZ 80 10

KdZ 12 6

µZ 0.9

λZ 0.4

Developing an identical analysis, Figures 16 and 17 and Table 2 show the time response of the system.

x-velocity [m/s]

6. CONTROLLER PERFORMANCES

time [seconds]

Fig. 17. Time response of the system under the action of the PIλDμ controller with Gradient Optimization. Table 2. Time response parameters of the system under the action of the PID and PIλDμ controllers with Gradient Optimization PO(%) tr tp ts 23.29 21.58 22.58 31.02 PID PIλDμ 19.04 21.28 24.26 29.00 We can clearly see the advantages of the FO algorithm that leads to a lower overshoot and a smaller settling time. Based on the previous optimized parameters we will run a PSO algorithm that will determine the optimal parameters of the controller, in order to obtain the minimum error when the bird moves between two workspace points. In the experiments it is adopted i max = 1.6 and 6 particle population during 100 iterations (Table 3). Table 3. PID and PIλDμ controller parameters with PSO Optimization

x-velocity [m/s]

PID PIλDμ

KpX 10.8 20

KiX 1.3 1.1

KdX 10.4 8.1

µX 0.86

λX 0.9

KpZ 54.3 25.6

KiZ 14.5 10.2

KdZ 15.3 5.6

µZ 0.9

Developing an identical analysis, Figures 18 and 19 and Table 4 show the time response of the system.

time [seconds]

Fig. 16. Time response of the system under the action of the PID controller with Gradient Optimization.

λZ 0.6

x-velocity [m/s]

It is possible to simulate all kind of closed loop actions like gliding, flapping wings, taking off, landing, following trajectories and others. Information relative to the physical nature of the flapping flight proved to be important to analyze solutions. The results had been evaluated using an intuitive analysis, and equally validated by other preceding analysis in this area. For this system are compared integer and fractional algorithms. The results of the fractional controller reveal better performances than the classical integer order controller.

REFERENCES time [seconds]

x-velocity [m/s]

Fig. 18. Time response of the system under the action of the PID controller with PSO Optimization.

time [seconds]

Fig. 19. Time response of the system under the action of the PIλDμ controller with PSO Optimization. Table 4. Time response parameters of the system under the action of the PID and PIλDμ controllers with PSO Optimization PO(%) tr tp ts 22.28 21.20 23.68 27.20 PID PIλDμ 21.40 21.16 23.74 27.04 The PSO optimization shows, in general, better results than the Gradient Descent optimization. Better results could be obtained increasing the population and/or the number of iterations. On the other hand it would significantly increase the simulation time. 7. CONCLUSION We obtained some results that appeared to be satisfactory which proves that the development of the kinematical and dynamic model can show us the behavior of the bird.

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