Shape Optimization of a Curved Beam Hopping Robot

Transcription

1 Prof. Dr. Fumiya Iida Master-Thesis Shape Optimization of a Curved Beam Hopping Robot Autumn Term 2012 Supervised by: Nandan Maheshwari Xiaoxiang Yu Murat Reis Author: Dominik Naef

2

3 Acknowledgments I would like to express the deepest appreciation to Prof. Fumiya Iida who enabled me to work independently and to suggest and fullfil my own ideas. He always found time for a constructive discussion and was an important motivater. I would also like to thank my suppervisers Nandan Maheshwari, Xiaoxiang Yu and Murat Reis for their patient, inspirational and proficient support throughout my entire work. This thesis is dedicated to my parents Regula and Thomas Näf for their endless love, support and encouragment. i

4 Contents Abstract Symbols v vii 1 Introduction 1 2 The Curved Beam Hopping Robot The main idea of a Curved Beam Hopping Robot Free vibration modes of a curved beam Hopping principle The design of the robot Curved Beam Foot Additional weight Actuation The Simulation SimMechanics Existing Simulations The model General design The foot and the Ground Contact Model The curved beam Actuation and control Visualization Solver Outputs of the simulation Parameters Problems during the implementation of the simulation System Identification The choice of a criterion to fit Data observation in the real world The selection of a suitable model The parameter estimation The model validation Summary of the system identification The Optimization The optimization problem The Genetic algorithm Introduction to the Genetic Algorithm Implementation in MATLAB ii

5 5.3 Experiments Methodology to show the increase of the performance Methodology to show the repeatability of the results Methodology to validate the optimization results Problems during the implementation of the optimization Simulation problems Real world problems Results Increase of the performance in the simulation Repeatability of the optimization results Validation of the optimization in the real world Analysis of the results Analysis of the increase of the performance in the simulation Analysis of the results of the repeatability Analysis of the optimization results in the real world Conclusion 43 9 Future Work 44 A The SimMechanics Simulation 45 A.1 The model hierarchy A.2 Model overview A.3 Actuation A.4 Animation A.5 Curved Beam A.6 Ground Contact model A.7 DC-motor model B The parameter-file 55 C OptiTrack data Handling 57 D Parameter estimation with a fmincon algorithm 61 E The optimization function 63 Bibliography 67 iii

6 iv

7 Abstract A curved beam hopping robot makes use of elastic and energy conserving materials by stimulating the different free vibration modes of a curved beam. This stimulation can result in different deflection patterns and therefore in different walking and running gaits. Since these dynamic robots are not trivial to understand and design, this work offers an approach on how to obtain an optimal shape of a curved beam hopping robot. In order to do so, a model of the robot and its environment was created in MATLAB SimMechanics. The simulation was validated by showing that the same resonance frequencies can be found in the simulation, as well as in the real world experiments, and the hopping behavior looked very much alike. In a next step, the shape of the robot was optimized in the simulation with a genetic algorithm. It could be shown in the real world that the performance of the robot was increased significantly and that this design approach offers a promising alternative to existing approaches. v

8 vi

9 Symbols Symbols ζ L ζ T ω M B L B B B I B L LegN L Orig α m LegN I LegN k LegN d LegN k T d T M T I T M R L R M RArm Longitudinal resonance frequency Torsional resonance frequency Actuation speed The weight of the foot The length of the foot The width of the foot Inertial of the foot calculated from the mass and the dimensions Length of an increment of the curved beam Origin of the curved beam relative to the CG of the foot in x-direction Angels between the increments of the curved beam, depending on the shape Mass of an increment Inertia of an increment Spring stiffness between the increments Damping rate between the increments Spring stiffness between the foot and the beam Damping rate between the foot and the beam Mass of the increment at point T Inertia of the increment at point T Rotating mass Radius of the rotating mass Mass of the arm of the rotating mass µ slide Sliding friction coefficient µ stick Static friction coefficient v th Velocity threshold for the JSA k z Spring constant for ground reaction force d z Damping rate for ground reaction force k y Spring constant for ground reaction force d y Damping rate for ground reaction force g Gravity vector Acronyms and Abbreviations ETH Eidgenössische Technische Hochschule vii

10 CBHR VRML CG CS GCM JSA FFT GA Curved Beam Hopping robot Virtual Reality Modeling Language Center of Gravity Coordinate system Ground contact model Joint-Stiction-Actuator Fast Fourier Transformation Genetic Algorithm viii

11 Chapter 1 Introduction In recent years, energy efficiency played a bigger and bigger role in the context of robotic walking and hopping locomotion. In nature it can be seen, that high energy efficiency can only be achieved by the exploitation of elastic components, such as tendons in animals. In the world of mechanics this can be achieved by using mechanical springs or other elastic forms such as rubber bands or curved beams. The latter is used in a new approach which is studied mainly at the Bio-Inspired Robotics Lab. A Curved Beam Hopping Robot (CBHR) consists of an aluminum (or other elastic material) beam, which is actuated by an electric DC-motor with a rotating mass (see figure 2.1a). When actuated at the right frequency, the free vibration modes of the curved beam can induce a stable forward locomotion. The advantages of this type of robots are various. First, due to the capability to recuperate the energy at impact, the low damping rate of the curved beam and the low degree of actuation, these robots are very energy efficient. In [1] it is shown that a specific resistance of down to 0.2 can be achieved using the right settings, which is very good for a hopping robot. Other great advantages are its low material costs and their small technical requirements in the construction. A CBHR can be built within one day and at very low costs. On the other hand there re also some drawbacks: Since the hopping behaviors of these robots depend on a wide range of different parameters (such as shape of the curved beam, size of the foot, mass of the foot, mass of the curved beam, rotating mass, rotating speed, et cetera), they are still not fully understood. A better knowledge is crucial mainly for increasing the controllability of these robots and for creating a more straightforward design strategy. The shape of the curved beam - as a major factor influencing the performance of the robot - cannot be determined in an analytical way. This work focuses on the introduction of a new approach to design the shape of the curved beam in an optimal way; for a discussion of the other design parameters please revere to [1], [2] and [3]. The design approach presented in this thesis is based on a SimMechanics simulation of the hopping robot, which is optimized by a Genetic Algorithm. Previous studies ([1]) described the shape of the curved beam with the two Parameters d h and L 0 and the curvature ratio e = d H /L 0, as shown in figure 2.1b. This simplification allows us to make basic predictions of the hopping behavior of variably shaped curved beams. It enables us also to build a simple simulation of the hopping robot. However, it does not provide enough information to make a profound statement about the behavior of the robot or even to build the curved beam based on this prediction. For a more detailed description of the shape 1

12 Chapter 1. Introduction 2 a more sophisticated model is needed. A possible approach is shown in this work in chapter 3. The model used in the simulation is optimized at a certain actuation speed by using a Genetic Algorithm. This work is structured as follows: In Chapter 2, a brief description of the CBHR is provided and its basic features are mentioned. In Chapter 3 the model of our robot - as used in the simulation - and its key parameters are introduced. The approach and the results for the system identification are provided in Chapter 4. In Chapter 5 the optimization approach is presented and different factors that influence the shape of the curved beam are offered. In Chapter 6, the results are presented. The work is summed up with a brief discussion and a conclusion in the Chapters 7 and 8.

13 Chapter 2 The Curved Beam Hopping Robot 2.1 The main idea of a Curved Beam Hopping Robot To be able to optimize the shape of the curved beam of a Curved Beam Hopping robot, first, the robot itself had to be examined and understood. For that matter a robot was built in the real world. The main idea of creating a CBHR (as shown in 2.1a) is to build a very energy efficient hopping robot. This goal is reached by using the free vibration modes of a curved beam in a way that they amplify a motion induced by a rotating mass around the top point T of the beam (see figure 2.1b). Ideally, the actuation stimulates the curved beam in a way that induces an Up-And-Down motion of the top point T. The Up-And-Down motion is created with the help of the centripetal forces of a rotating mass. To make sure that these centripetal forces induce a Up-And-Down motion it is crucial to understand the importance of the free vibration modes of the curved beam Free vibration modes of a curved beam The two most important vibration modes are the torsional oscillation mode and the longitudinal oscillation mode. For investigating these two oscillation modes, the real world robot was fixed to the ground and actuated at different rotational speeds. The torsional oscillation mode occurred at a motor speed of approximately 7 rad/s and shows a back and forth rocking of the point T. The second, longitudinal oscillation mode occurred at around 19 rad/s and shows a Up-And-Down motion of the point T. The motion of the point T at the two resonance frequencies is shown in figure 2.2. It can be seen that the motion of the point T is strongly dependent on the actuation speed. Even though these frequencies can be found easily in the real world experiments, the analytical determination is very difficult since the correlation of the parameters is very high. In [1] it is shown, that the main parameters influencing the resonance frequencies include the mass at the top M T, the length L 0 and the stiffness of the beam. The latter two are strongly affected by the shape of the beam. To achieve a stable forward hopping - in our case - mainly the second oscillation mode is used. 3

14 Chapter 2. The Curved Beam Hopping Robot 4 (a) Photograph of the Curved Beam Hopping Robot used. (b) Abstracted Robot with key parameters denoted. Figure 2.1: Two representations of the robot used in this work. Figure 2.2: Torsional and longitudinal oscillation of point T at resonance frequencies one and two.

15 The main idea of a Curved Beam Hopping Robot Figure 2.3: The different phases of a Curved Beam Hopping Robot when hopping Hopping principle When the Up-And-Down motion is strong enough, the robots foot is able to lift off from the ground so that the locomotion can be divided in a flight and a stance phase. To get the robot airborne, the centripetal force in vertical direction and the force exerted by the energy stored in the curved beam added together must be greater than the weight force of the whole robot: F cent. + F Beam > G M where F cent. is the centripetal force, F Beam is the force exerted by the beam and G M is the gravity force of the whole robot. The centripetal force is calculated as F cent. = M R L R ω 2 with M R as the rotating mass, L R the radius of the rotating mass and ω as the rotational speed. The force F Beam can be approximated by a spring force, to get an idea of the amplitudes: F Beam = k Beam L where k Beam is the approximated spring stiffness and L is the deflection from the original spring length. This approximation is only used here to get a basic understanding of the phenomena appearing. For further discussion please refer to [1]. Since in the creation of our robot the rotational mass is kept as low as possible, the robot can only hop when the rotational speed of the rotating mass and the longitudinal free vibration mode coincide, as shown in figure 2.3. In the figure, the solid line shows the curved beam in action and the dashed line shows the curved beam in its initial position. The green arrows represent the forces acting upon the point T and the red arrows show the velocities of the point T and the point B. The rotating mass rotates clockwise. 1. In the first phase, the centripetal force is pointing downwards, which causes the point T to move downwards and therefore to tension the curved beam. The energy that would have been lost without an elastic element is stored as potential energy in the beam.

16 Chapter 2. The Curved Beam Hopping Robot 6 2. After point T has reached its lowest position, it starts to move upwards again. At the same time the centripetal force starts to point up and the energy stored in the beam is released by inducing an ascending force on point T as well. 3. In the third phase, the robot gets airborne. The centripetal force and the energy stored in the curved beam accelerated the point T and the curved beam in an extend, that allows the robot to lift off. Even though the force coming from the beam gets negative, the still acting centripetal force and the inertia of the bodies in motion are enough to cause this phenomena to happen. 4. In the fourth phase, the robot has reached its apex height and starts to descend again until it reaches the ground. Thanks to the elastic structure of the curved beam, the kinetic energy is not lost at touch down. Instead it is to a great part converted back to potential energy. 2.2 The design of the robot In figure 2.1a the robot used in this work is presented. As can be seen, it is a very simple design consisting only of 4 major parts: The foot, the curved beam, the DC-motor and some additional weight at the top. The design is based on previous studies which can be found in [1] and [3]. For reasons of reusability and to facilitate the assembly of the robot, the individual parts were held together with hot melt adhesive (HMA). More details on the design of the individual parts can be found in the following subsections Curved Beam The most difficult part of the design is the curved beam. Since the behavior of the curved beam and the exact influence of its design parameters is not fully understood, no reliable design strategy is existing so far. That s why a new approach is presented in this work. Nevertheless, some basic statements on its design can be made. Since a great part of the mass of the robot is located at point T, it is important to make sure that this point lays above the foot stance of the robot in its initial condition. Otherwise the system flips over immediately. Since the curved beam is highly elastic, it is important that this point can t move to far out of this comfort zone, because the stability of the robot in dynamic conditions is still very limited. As well as the horizontal position, the vertical position of point T is important. On one hand, the higher this point lays the less stable the robot is. On the other hand, a higher position of this point benefits higher hopping heights. Already very small changes of this position can lead to big differences in the hopping behavior concerning the hopping speed, hopping height but also the hopping stability. The shape of the curved beam itself should be chosen in a way that the first vibration modes are clearly distinguishable. Depending on how the shape is chosen, the frequencies where the free vibration modes lay can be influenced strongly. This will be shown in Chapter 6. The curved beam was made of aluminum, had a total length of m and a weight of kg. To be able to design the beam and to quantify the shape in a reliable way, first,

17 The design of the robot Figure 2.4: The shape of a Curved Beam Hopping robot represented with 5 increments and 5 angles. a meaningful representation of the shape is needed. In earlier studies, the simplification with the parameter e = d H /L 0 was used, consisting of the parameters d h and L 0 as mentioned in Chapter 1. For our purpose a more sophisticated depiction was needed, since even curved beams with same values for e, d h and L 0 can have their free vibration modes at completely different frequencies. Here fore, the curved beam was divided in several increments of given length L inc. The angles between each pair of increments are used to describe the shape. The length of the increments are chosen in a way that all the increments added up are equal to the length of the curved beam. L inc = L Beam /N inc In figure 2.4 a two dimensional example with four increments and the according angles α 0 to α 4 is shown. In our work, only two dimensional representation is used. With this representation we were able to describe the shape of the curved beam in as much detail as needed. By only using two dimensional shapes, it was possible to obtain the shape from a photograph with relatively small effort. The more increments that are chosen, the more detailed is the depiction, yet the more additional expenditure was needed to get the angles Foot The design of the foot was very straightforward. The goal was to get the foot stance as wide as possible but at the same time keep the weight as low as possible. This was achieved by a combination of wood and aluminum. The final dimensions were a foot stance of m times m at a total weight of kg.

18 Chapter 2. The Curved Beam Hopping Robot Additional weight Besides the curved beam itself, the weight of the point T on top of the curved beam was the most important factor influencing the resonance frequencies of the curved beam. Previous work ([1]) showed that the heavier the mass at the top, the lower the resonance frequencies. A lower resonance frequency is more advantageous for a higher hopping height and therefore a faster hopping speed, while at the same time causes a more stable locomotion of the robot. On the other hand it adds additional weight to the robot, so again a compromise had to be found. In our robot we chose to put the additional weight of one battery at the top of the robot, also to show that the robot could function without an external power source. The total weight of the gearbox, the DC-motor and the additional weight added up to kg Actuation For the actuation of our model a simple FA130-RA DC-motor was used in combination with a Tamiya gearbox with a gear ratio of 115:1. The gear ratio was chosen that high to get the rotational speed of the rotating mass as constant as possible. Another possibility to achieve this would be to take a superior DC-motor link with an encoder. Even though this option was considered, it would have been too complicated and too expensive for our purpose. The DC-motor was linked to an external voltage source with no further controllers. For adjusting the rotational speed, the voltage was regulated manually, based on a speed-voltage curve which showed the relationship between these two values.

19 Chapter 3 The Simulation To be able to optimize the shape of the curved beam, a realistic simulation of the hopping robot is needed. This simulation has to fulfill some basic requirements: First of all, it has to represent the real world robot - especially the curved beam - in enough detail to be able to design the real world robot based on the results of the optimization. Secondly, the representation should be accurate. Since already in the real world, very small changes in the design can lead to big differences in the hopping behavior, these effects should be reproducible in the simulation as well. The third, very important aspect is the computational effort. In the case of an optimization, the simulation will be run many times. Therefore, a high computational time is disadvantage. Since this model is intended to be used beyond the studies of this thesis, the simulation should be structured in an understandable way. To be able to achieve the criteria mentioned above, we choose to use MATLAB SimMechanics as our simulation environment. With MATLAB SimMechanics we are able to build the model in as much detail as needed and it is easily understandable. Furthermore, the computational effort is within a reasonable frame and the possibility to use MATLAB as an environment for the optimization is a great advantage. 3.1 SimMechanics MATLAB SimMechanics is a Toolbox within the MATLAB Simulink environment. Whereas Simulink is built mainly to model, simulate and analyze multi domain dynamic systems in general, SimMechanics is explicitly developed to simulate mechanical systems. These mechanical systems consist of rigid bodies connected by joints, with the standard Newtonian dynamics of forces and torques. Whereas in Simulink the individual blocks represent mathematical operations or operate on signals, a SimMechanic block represents a physical component or relationship directly. Like that we can build complex mechanical systems, without the need of forming the equations of motion first. To combine Simulink and SimMechanics models, special blocks are needed to feed the signal from one domain to the other. These blocks are either actuation blocks 9

20 Chapter 3. The Simulation 10 Figure 3.1: The previous simulation and its visualizations. (from Simulink to SimMechanics) or sensor blocks (from SimMechanics to Simulink). When sticking to this simple restriction, any Simulink block (including embedded MATLAB functions) can be used in combination with SimMechanics. Exactly as in Simulink, it is possible to model systems with hierarchical subsystems, define parameters externally and visualize the simulation with the help of the Virtual Reality Modeling Language (VRML) (see subsection 3.3.5). In the SimMechanics toolbox, simple blocks for bodies, joints, sensors and actuators, constraints and drivers and force elements are included. Besides these simple blocks there are also blocks with advanced functionality available, such as the Joint- Stiction-Actuator which will be discussed in more detail in subsection The SimMechanics toolbox is only one part of the physical modeling toolbox called Sim- Scape, but in this work we will be focusing solely on the mechanical modeling so it will be the only one used here. An exception forms the modeling of the DC-motor which is discussed briefly in the subsection For additional information about MATLAB, Simlink and the SimMechanics toolbox please consult [4] and [5]. 3.2 Existing Simulations Since these hopping robots have already been studied intensively at the Bio-Inspired Robotics Lab, there were already some simulations existing. The model used in this work is based on a previous simulation implemented by M. Reis. This model was also build in MATLAB SimMechanics and simplified the curved beam as a combination of a torsion and a linear spring (see [1]). It was actuated with a rotating mass rotating at a constant speed and was modeled 2 dimensional. In figure 3.1 you can see the visualization of the robot in SimMechanics and in a separate hand build visualization.

21 The model Even though this simulation delivered good results - once the right parameters were found - there were several reasons why the implementation of a new simulation was inevitable: The curved beam was simplified by a linear and a torsion spring. To make an expressive prediction on the shape of the curved beam, this simplification is not suitable. The model is simulated in 2 dimensions only. The ground contact model is not modeled in great detail. It allows the foot to penetrate the ground too extensively. The inertias of the rigid bodies used is not realistic, since the rigid bodies don t have similar shapes as the structures in the real world. The visualization could be done more attractively. 3.3 The model General design To model the Curved Beam Hopping robot in SimMechanics a 3 dimensional approach was chosen. The real world robot should be represented as good as possible, that s why for every major element of the real world robot, at least one rigid body block was implemented with the same dimension, mass and inertia. The model consisted mainly of six parts. The foot and the ground contact model, the curved beam, the actuation, the animation, the visualization and an external parameters file. The blocks for defining the output of the model and the blocks for the simulation environment and the solver options were implemented as well. An overview of the model can be seen in figure 3.2. More detailed descriptions of the individual parts follow in the next subsections The foot and the Ground Contact Model The first part described here is the foot of the robot and its ground contact model (GCM). Unlike in the real world, in the model the foot is represented as one rigid body. The foot can be assumed as a rigid body since it doesn t contribute to the hopping of the overall robot. It has the same dimensions as the foot in the real world, the same mass and the same inertia. Body Block: In figure 3.2 the foot is modeled by the red body called Foot. By doubleklicking on this block a graphical interface appears where all the options concerning this body can be defined (see figure 3.3). A brief description of these options follows, representative for all the following body blocks: As can be seen there are first the entries for the mass and the inertia of the body which are trivial. The next entries describe the ports that connect the foot to its environment: CG: The block with the name CG describes the location of the Center of Gravity of this body block. You can see that the CG is not displaced relative to the world, that means that the robots initial condition is at the origin of the world coordinate system. As you can see in figure 3.2 this port is connected to the ground and the machine environment. The ground block is necessary to connect the model of the robot to its environment. In the machine

22 Chapter 3. The Simulation 12 Figure 3.2: An overview of the SimMechanics simulation.

23 The model Figure 3.3: Body block options shown at the example of the robots foot. environment block the settings of the environment and the constraints are set (see subsection 3.3.6). CS5 to CS12: The coordinate systems 5 to 12 denote the 4 corners of the robots foot. The positions are described relative to the CG of the foot, which lays in the middle of the foot. These 8 ports are connected to the ground contact model (GCM) where the ground reaction forces are calculated and exerted on the body. CS13: The coordinate system 13 is linked to the curved beam. The displacement relative to the CG shows that the curved beam is not mounted in the center of the foot, but with a relative displacement of L Orig. CS14: The coordinate system 14 is again placed in the CG and is used to measure the position of the foot relative to the world coordinate system in order to visualize the robot (see subsection 3.3.5). The options concerning the orientation are discussed in the subsection Ground contact model: For calculating the ground contact forces the two green subsystems called Ground Interaction Model were created. To minimize the computational effort the ground contact forces were calculated in each of the three dimensions in different ways. In z-direction - which is the vertical direction - a soft contact model was chosen with a linear spring damper system. The advantage of a compliant model against a hard contact model is that in a hard contact model a geometric constraint is used (i.e. the foot can t penetrate the ground), which causes the system to lose a degree of freedom. In the soft contact model on the other hand there is just an additional force acting upon the foot, but the equations of motion basically stay the same. On top of that a soft contact model as used in this work allows the body to bounce

24 Chapter 3. The Simulation 14 back from the ground, like in a certain degree is also happening in the real world. For a deeper discussion of the different types of GCM, please refer to [6], [7], [8], [9], [10] and [11]. The spring damper system used here was of the form { k z z d z ż if k z z d z ż 0 f z = 0 else (3.1) and was only applied for the case z 0. (3.2) f z denotes the force in vertical direction, k z is the spring stiffness and d z the damping rate. z represents the vertical position of the foot point and ż the vertical velocity. As can be seen, only forces acting in positive z-direction are allowed, since we want to avoid sticking forces which cause the model to stick to the ground. The advantage of this type of equation is its simplicity (only two additional parameters have to be defined) and its good representation of the reality. On the other hand the differential equation resulting from this method are very stiff and therefore very time consuming to be calculated. On top of that, the forces acting upon the foot in the moment of the touchdown are not continuous, which is not a good representation of the reality. The calculation of these forces were implemented by conventional MATLAB Simulink blocks. To measure the position of the robots foot, a body sensor block from the SimMechanics library was used. To exert the force on the foot, an actuator block was taken. An overview of the implementation of the GCM can be found in the Appendix A.6. In y-direction, a very similar, simple approach was chosen since this isn t the main locomotion direction. Again a linear spring damper system of the form was implemented as can be seen in the Appendix A.6. f y = k y y d y ẏ (3.3) In x-direction a more elaborate approach has to be chosen since simpling and stiction is a fundamental part of the forward locomotion of the robot. To represent these two phenomena we have to be able to either apply a zero velocity constraint on the joint in case of stiction, or apply a friction force in case of slipping. Unfortunately in MATLAB SimMechanics it is not possible to actuate one degree of freedom with a speed driven approach and with a force driven approach at the same time, since redundant inputs would be the result. To solve this problem MATLAB provides a block with advanced functionality, the so called Joint-Stiction-Actuator (JSA). The JSA applies a friction force upon the joint primitive as long as the joint is unlocked. In the case of a locked joint, a zero velocity constraint is applied. To distinguish between these two actions, the JSA has three discrete joint modes, unlocked, locked and wait. In the locked mode, the joint primitive gets restricted by a zero velocity constraint and cannot move at all. This mode is activated as soon as the velocity is smaller than a certain velocity threshold v th.

25 The model Figure 3.4: Joint stiction modes and transition condition from the MATLAB documentation. In the unlocked mode the joint primitive moves with a kinetic friction applied. To unlock a locked primitive first the wait mode must be activated. In the wait mode the joint primitive is still locked, but static computed force in the joint exceeded the static friction threshold F s. To make the transition from the wait mode to unlock mode, a virtual velocity is calculated to predict the movement in case of an unlocked joint. If this velocity lays above a certain threshold v th the joint translates to unlock mode. A schematic visualization of these state transitions can be seen in figure 3.4. The implementation of the GCM in MATLAB SimMechanics can be found in the Appendix A The curved beam The representation of the curved beam in the simulation was a fundamental part of this work, since this was the component we wanted to optimize. The approach chosen was to divide the curved beam into a number of rigid bodies connected with torsion spring damper systems. The beam was split into 16 rigid bodies, as is illustrated in figure 3.5. Between each pair of rigid bodies a torsion spring damper system was implemented with a rest position that corresponded to the angle between the two increments in the initial shape. The angles were defined as the rotation of one rigid body relative to its predecessor, as pictured in figure 2.4. In figure 3.6 you can see two subsequent increments of the beam. As can be seen one increment consists of three standard blocks from the SimMechanics library. In the revolute block it was specified which degree of freedom is activated. Here only the rotation along the y-axis is possible, since we want the curved beam to be elastic along this axis. In the joint spring and damper block only the spring constant k leg and the damping coefficient d leg must be defined. In the body block the mass and the inertia of the body were defined. On top of that, the location of the revolute

26 Chapter 3. The Simulation 16 Figure 3.5: Abstraction of the curved beam. Figure 3.6: Two subsequent elements of the curved beam. joints must be specified by introducing additional coordinate systems. By doing so, the dimensions of the increments and the initial angle between the two bodies were defined as well. The relative rotation between the coordinate systems was defined by the rotation vector and the angle α n along this vector. We used the Euler X- Y-Z convention, all though we only used one rotational direction. The graphical interface to do so is pictured in figure 3.7. The angles between the increments were named α 1 to α 15. The angle between the foot and the first increment of the curved beam was named α 0. The spring constant k leg and the damping rate d leg was set the same for all the spring damper systems in the curved beam. The spring constant and the damping rate between the foot and the first element of the curved beam were named k T and d T and were allowed Figure 3.7: Body block options shown at the example of a beam increment.

27 The model to have a different value than the rest of the curved beam to increase the flexibility in the System Identification. All the increments had the same length, mass and inertia for the sake of simplicity and due to the fact that the beam has a uniform stiffness. The number of 16 beam elements was chosen as a compromise between model accuracy and computational effort Actuation and control For the actuation of the rotating mass three different options were existing: A constant speed control, a torque control and the control via a modeled DC-motor. In the following the three options were introduced and their pros and cons mentioned. Speed control. With this approach the rotating mass is always actuated with a constant angular velocity. This approach has various advantages: It is very simple to implement and it is easy to measure and control. It is relatively accurate since in the real world the motor almost delivers a constant speed when connected to an ideal power source. Previous studies showed that the oscillation of the angular velocity in the real world is within a reasonable frame. Torque control. With this approach the rotating mass is actuated at an always constant torque. Again this method is very simple to implement, but it s not as simple to control. On top of that it does not reflect the reality in a good way, since a DC-motor does not deliver a constant torque at all. DC-motor model. With this approach the motor was modeled as a DC-motor (which was given in the MATLAB SimElectronics library) and some additional blocks to convert the signal from a SimDriveline signal to a SimMechanics signal. (The DC-motor model can be found in the Appendix A.7.) This approach would have been the most realistic representation of the real world since all the DC-motor specification and friction parameters could be defined. But since the DC-motors we used in the real world a very cheap, the specification sheet of these engines was not very accurate. The internal parameters of the DC-motor would have needed to be found experimentally which would have gone beyond the scope of this thesis. On top of that the DC-motor model turned out to be quite computationally expensive. Due to the reasons mentioned above, we decided to actuate our model with a constant speed control approach Visualization The MATLAB SimMechanics environment offers three basic ways to visualize a simulation: A hand made animation based on the coordinates of the system (as used in the already existing simulation), the built in animation with only very few options and poor graphics and the integration of a Virtual Reality Modeling Language (VRML) world. Due to the better performance and the more elaborate possibilities the approach via VRML toolbox was chosen. VRML in MATLAB Simulink VRML is a standard file format for the visualization of 3 dimensional vector graphics. Even though it was mainly designed for the world wide web it is also widely used to visualize and verify dynamics systems in scientific projects. The animation is done by changing object properties such as position, rotation, shape and scale

28 Chapter 3. The Simulation 18 Figure 3.8: The model visualized in VRML. during desktop or real-time simulation. The MATLAB Simulink 3D Animation toolbox offers Simulink blocks to connect the virtual world with the model, a 3D World Editor with a graphical interface to the VRML syntax, a video recording function and a lot more. With the 3D World Editor VRML nodes can be assembled which are connected to the Simulink environment. These nodes can specify aspects such as appearance, navigation, geometry, groups, interpolators, lights and sensors. For our use, mainly the geometry and the groups were interesting. In a Transform group all the information about orientation, translation and scale could be handed over from Simulink. With the geometry nodes the dimensions of the body parts visualized can be defined. How was it used in the simulation In the 3D World Editor a virtual world was created consisting of geometry parts for all the bodies defined in the SimMechanics model. These nodes were connected with the model using a VR Sink block. In this block all the nodes of the virtual world created can be seen in the VRML Tree, and the nodes feeded with information from the model can be chosen. In our case the information handed over were the dimensions of the bodies, the orientation and the translation. Furthermore a sample time could be defined here, which was set to 0.01 s. The dimensions of the bodies were calculated directly from the parameters from the parameter-file (see subsection 3.3.8). The information concerning the orientation and translation were gained with the use of Body Sensor blocks and handed over with the help of Go To and From blocks. A picture of the robot visualized in the VRML world can be seen in figure 3.8.

29 The model Figure 3.9: Solver hierarchy in a SimMechanics simulation from [4] Solver To run the simulation successfully the machine and the model-wide settings had to be set. The model-wide settings were set by the definition of the settings of the Simulink Solver which controls the purely mathematical aspects of the simulation. The SimMechanics software interprets the machine s purely mechanical aspects through machine assembly and a constraint solver. The two parts were combined in a dynamic hierarchy, as can be seen in figure 3.9. The machine settings are defined for each machine of the model individually and contain the mechanical settings of the machine. They were implemented by the use of a Machine Environment block and are SimMechanics specific. Here the simulation dynamics, machine dimensionality, gravity, tolerance, constraints, motion analysis modes and visualization settings can be set. Following setting were chosen.

30 Chapter 3. The Simulation 20 Gravity vector [ ] Gravity acts in negative z- direction. Machine dimensionality 3D Only Analysis mode Forward dynamics We want to calculate the positions and velocities of the bodies continuously based on the forces acting upon them. Linear assembly tolerance 1e-3 Angular assembly tolerance 1e-3 Constraint solver type Tolerancing Model stability is more important than accuracy. Relative tolerance 1e-4 Absolute tolerance 1e-4 Redundant constraint Specify tolerance tolerancing Relative tolerance 1e-4 State perturbation Adaptive type Perturbation size 1e-5 The model-wide settings are defined in the Simulink Configuration Parameter dialog. The Solver options were set to: Type Variable-step When the ground contact model is active a much smaller Time step is needed than in the flight phase. Solver ODE45 (Dormand- Prince) A relatively fast solver also for stiff differential equations. Relative tolerance 1e-4 Max step size auto Min step size 1e-4 Smaller step sizes can lead to a very long simulation time. Absolute tolerance Auto Initial step size Auto Shape preservation Disable all Enabling time preservation is computationally expensive. As can be seen, the settings were chosen in a way to get a fast and stable simulation. The model accuracy was not the major concern Outputs of the simulation To be able to optimize the simulation with an external MATLAB function, we had to specify which state variables were chosen as output. Since we were mainly interested in the hopping performance of the robot and not on its controllability or orientation, it was enough to hand over the position vectors of some points of interest. With the use of To File blocks we wrote out the positions of the CG of the foot, the position of the point T and the CG of the rotating mass every 1/120 of a second in the world coordinate system. The data was stored in the files called Trackable1, Trackable2 and Trackable3.mat.

31 Problems during the implementation of the simulation Parameters To structure the model is understandable and modifiable as possible, all the parameters used to define the body properties were specified in a separate m-file. The values can be adjusted easily in this file and exported to the basic workspace, from where the simulation could read them in. A full list of the parameters used can be found in the Appendix B. Their determination is discussed in the Chapter Problems during the implementation of the simulation During the implementation of the model several bigger problems occurred. Following a brief discussion of the main problems: The GCM was very tricky to implement to work stable. The loss of a degree of freedom in the case of a locked joint causes the simulation to crash likely. The problem could be solved by choosing higher assembly tolerances and an ODE45 solver. With other solvers, e.g. a ODE15s which would be better fitted for solving stiff differential equations, the simulation does not work. The DC-motor simulation was very tricky to implement since different MAT- LAB toolboxes had to be combined. The right settings for the signal conversion were not at all trivial to find. In general it can be said that the smallest mistakes in the model can cause the simulation to crash or to deliver completely wrong results. A wrong determination of the inertia of the rotating mass for example caused a lot of problems, since it is really hard to spot the bug. The most difficult part in the implementation was the determination of the parameters used for the model. This determination will be discussed in the next chapter.

32 Chapter 4 System Identification Once the model was created in MATLAB SimMechanics a system identification approach was needed to find the best fitted parameters for the model. Since the robot is build very similar compared to the real world, many parameters, such as the dimensions or the masses of the individual bodies, could be determined experimentally, ergo no system identification was performed for these values. The system identification process mentioned below was necessary for the determination of the spring constants and the damping rates of the curved beam k LegN, d LegN, k T and d T. In general, the system identification procedure is characterized by five basic steps: The choice of a criterion to fit. The data observation in the real world. The selection of a suitable model. The parameter estimation. The model validation. The following sections will discuss the five steps mentioned adopted to our problem. 4.1 The choice of a criterion to fit To determine the parameters mentioned above, we wanted to find the same behavior in the model as in the real world robot. Since the hopping behavior is strongly depending on the occurrence of the free vibration modes, we implemented our system identification process based on this phenomena. The frequencies of the free vibration modes can be found easily in the real world robot. The goal was to show that the same free vibration modes occur in the simulation and to define the parameters in a way that they appear at the same frequencies as in the real world. Like that we are able to find the best suitable parameters for the stiffness of the curved beam. To get reliable results, it was crucial to define the parameters for the shape of the curved beam first. To define the damping rates of the curved beam, we compared the amplitudes of the oscillation of point T. We did this comparison at different frequencies, as can be seen in figure

33 Data observation in the real world 4.2 Data observation in the real world As a second step, a reference data set from the real world had to be collected. We wanted to gain data reflecting the behavior of the curved beam actuated at resonance frequencies and also at other frequencies. To do so the real world robot was fixed to the ground to facilitate experimental setup. We recorded data from the real world robot in motion with the help of a tracking system called OptiTrack. OptiTrack is an optical motion tracking system based on infrared cameras and passive markers. Multiple markers are placed on rigid bodies which allows tracking all 6 degrees of freedom. In our case we located 4 markers at the foot, 2 markers at the point T and 2 markers at the rotating mass. The tracked data is written every 1/120 of a second and is stored in a.csv-file. The.csv-file was converted to an Excel file which could be imported more easily to MATLAB. Once imported to MATLAB the data had to be structured and filtered, since the data included irregularities and glitches. The adaptation included following steps: First the data was restructured and the center of area of the rigid bodies was calculated. An array resulted where the first entry denoted the time and the entries 2 to 10 the x-, y- and z-coordinates of the trackables. In a second step the coordinates of the trackables were translated to the origin. This was done by subtracting the median value of the first two seconds of the recording. In a third step the components with the value NaN (Not-a-Number) were replaced by their forerunners. Next we got rid of unusually strange data. For that matter, each entry was compared to its forerunner and the difference was calculated. When the difference was above a certain threshold the entry was replaced by the value of the forerunner. In a last step a butterworth filter of order 10 was applied. The data-handling algorithm can be found in the Appendix C. In figure 4.1 the behavior of the point T when actuated at different frequencies is shown. To be able to make a statement about the resonance frequencies, a Fast-Fourier- Transformation was performed on the data. The frequencies found for our robot were at 1.3 Hz (8.2 rad/s motor speed) for the torsional oscillation and at 3.2 Hz (19.9 rad/s motor speed) for the longitudinal oscillation, as can be seen in figure The selection of a suitable model The model selected is described in full detail in Chapter 3. For the system identification, the model was fixed to the ground. 4.4 The parameter estimation The parameter estimation turned out to be the most time consuming part of the system identification. Two different approaches were tried out to get the best suitable parameters.

34 Chapter 4. System Identification Trajectory of point T at 8.2 rad/s Vertical position of point T over time at 8.2 rad/s 0.5 z [m] z [m] x [m] Trajectory of Point T at 19.9 rad/s time [s] Vertical position of point T over time at 19.9 rad/s 0.5 z [m] z [m] x [m] Trajectory of Point T at 10.4 rad/s time [s] Vertical position of point T over time at 10.4 rad/s 0.5 z [m] z [m] x [m] Trajectory of Point T at 12.5 rad/s time [s] Vertical position of point T over time at 12.5 rad/s 0.5 z [m] z [m] x [m] time [s] Figure 4.1: The oscillation of point T actuated at 8.2, 19.9, 10.4 and 12.5 rad/s. 25 FFT at Torsional Resonance Frequency 20 Amplitude Amplitude Frequency [/s] FFT at Longitudinal Resonance Frequency Frequency [/s] Figure 4.2: The results of the FFT performed at the torsional and longitudinal resonance frequency.

35 The model validation Frequencies of free vibration modes vs. spring stiffnesses Freq. [Hz] k_{legn}: 9.5 k_{t}: 12 Frequency: k T [Nm/rad] k LegN [Nm/rad] Figure 4.3: Correlation between the parameters k LegN and k T and the longitudinal free vibration mode. After a first brief estimation of the magnitude of the parameters, a minimization problem was created. Different approaches were tried with a nonlinear least-squares algorithm ( lsqnonlin ), unconstraint nonlinear optimization ( fminsearch ) and a constraint nonlinear optimization ( fmincon ). Even though slight enhancements could be noted - most of all with the fmincon algorithm - satisfying results could not be found. The main problems for that were the definition of the target functions and the high computational effort of the problem. An example where the fmincon algorithm was used can be seen in Appendix D. The parameters optimization by hand tuning turned out to be more straight forward and more efficient than the automated one. The resonance frequencies found in the real world were achieved by adjusting the values for k LegN and k T. This task was simplified by the fact that the parameter k T mainly influences the torsional resonance frequency and the parameter k LegN mainly influences the longitudinal resonance frequency. In figure 4.3 the correlations between the parameters k LegN and k T and the longitudinal resonance frequency are illustrated. Due to the limited amount of data points stored, the resolution of the FFT is relatively poor. Nevertheless, it can be seen that the face is a lot steeper along the k LegN -axis. The best suitable values found were k LegN = 9.5 and k T = 12 Nm/rad. To find the values of the parameters d LegN and d T, the amplitude of the oscillation of the point T was observed at different frequencies. In figure 4.4 the amplitudes at different motor speeds are shown. As can be seen, the amplitudes match pretty well, except when actuated close to the resonance frequencies. This might be due to the non-linearity of the elasticity of the beam in the real world. The best parameters found were d LegN = and k T = 0.03 Nm/rads. 4.5 The model validation The parameters found were validated by looking at the hopping model. The robot was simulated with the parameters found and the trajectories of the points of interest were compared with the real world robot. In figure 4.5 you can see the

36 Chapter 4. System Identification 26 Oscillation of point T actuated at 11.3 rad/s in real world 0.4 Oscillation of point T actuated at 11.3 rad/s in simulation 0.4 z [m] 0.3 z [m] time [s] time [s] 0.4 Oscillation of point T actuated at 13.2 rad/s Oscillation of point T actuated at 13.2 rad/s in simulation 0.4 z [m] 0.3 z [m] time [s] time [s] 0.4 Oscillation of point T actuated at 16.0 rad/s Oscillation of point T actuated at 16.0 rad/s in simulation 0.4 z [m] 0.3 z [m] time [s] time [s] Oscillation of point T actuated at 18.8 rad/s Oscillation of point T actuated at 18.8 rad/s in simulation z [m] 0.3 z [m] time [s] time [s] Figure 4.4: The oscillation in vertical direction of point T vs. time. On the left side the data from the real world robot, on the right side the results from the simulation. comparison of the two. On the top you can see the trajectory of the hopping robot in the real world and on the bottom you can see the trajectory obtained from the simulation. By comparing the two hopping behaviors, the ground contact parameters could be adjusted to complete the model. 4.6 Summary of the system identification From the results shown above you can see that the model created in this work correlates nicely with the real world robot. The behavior and the motion of the curved beam - especially of the point T - match very well. The hopping speed and the hopping height are similar, albeit not exactly the same. The differences can have multiple causes. One possible reason are inaccuracies in the ground contact model. Another one is the non-linear behavior of the curved beam in the real world robot, which is modeled by linear spring damper system. That latter would explain why the differences are bigger the greater the amplitudes of the oscillations are. Nevertheless, qualitatively the model matches very well and is sufficient for our purpose.

37 Summary of the system identification 0.05 Hopping robot actuated at 17 rad/s in real world z [m] x [m] 0.05 Hopping robot actuated at 17 rad/s in the simulation z [m] x [m] Figure 4.5: On the top: The hopping behavior of the robot in the real world. Below: The hopping behavior in the simulation.

38 Chapter 5 The Optimization As mentioned earlier, the hopping behavior of a curved beam hopping robot depends on a wide range of different factors. To simplify the problem, here only the optimization of the shape of the curved beam is investigated, meanwhile all the other parameters were set constant. The shape of the curved beam was chosen because it is a major part of the robot, but at the same time not fully understood yet. 5.1 The optimization problem The goal of our optimization is to change the angles α between the increments in a way that the performance of the curved beam increases. We start of at a set of predefined initial angles which already induce some hopping behavior and we end up with a better final shape of the curved beam. The performance of the curved beam can be described in many different ways, such as energy efficiency, stability, hopping height or hopping distance. In this work only the latter one is examined. The goal is to hop as far as possible within 10 seconds without falling over. The optimization problem was defined as follows. Objective function: Increase the hopping speed of the robot, i.e. hop as far as possible within 10 seconds. To reformulate it as a minimization problem: f(α) = a x 10s for i = (5.1) x 10s is the position of the CG of the foot of the robot after 10 seconds of hopping. a is some value greater than 2 to ensure that no negative objective function occurs and α stands for the angles α that define the shape of the curved beam. Boundaries: The values for the parameters were restricted by the lower and the upper bounds which were defined as 3 if i = 0 LBα i = 0.1 if 0 < i < 4 0 else (5.2) and 28

39 The Genetic algorithm 1 if i = 0 UBα i = 0.5 if 0 < i < else. LB stands for the lower bound and UB stands for the upper bound. (5.3) Inequality constraints: To make sure that the shapes found in the optimization are also feasible in the real world, an additional inequality constraint was defined: g 1 (α) > 0.1 (5.4) g 2 (α) < 1.2π (5.5) with. g 1 = g 2 = α i (5.6) 0<i< The Genetic algorithm Introduction to the Genetic Algorithm To solve the optimization problem a Genetic Algorithm is used. A Genetic Algorithm is a search heuristics, which imitates natural evolution by mutating a population of chromosomes over several different generations to match the fitness function as good as possible (see [12]). In our case the chromosomes are the angles α of the curved beam and the population is a set angles that define the shape of the curved beam. The fitness function is defined by an objective function. The procedure is the following: Figure 5.1: The process of a Genetic Algorithm.

40 Chapter 5. The Optimization Initialization: To start the algorithm, first an initial population must be set. In our case these are the initial angles of the curved beam and a random variation of them. The number of different samples in the population is called population size and has a great impact on the calculation time and the quality of the results. In our studies, a population size of 8 individual parameter sets turned out to be a good choice, since the calculation of the fitness function was very time consuming. 2. Evaluation: In the second step, all the shapes generated in the initial population are evaluated regarding their fitness. The fitness function used was f(α) = a x 10s where x 10s is the position of the point B after 10 seconds of simulation relative to its initial position. If the robot fell over or the simulation crashed the fitness value was set to f(α) = a + 2. Repeat: The following steps 3 to 5 are repeated until either the fitness value reaches a certain threshold, or the maximal number of generations is reached. Since our goal was to hop as far as possible, no threshold was set for the fitness, but the number of generations was restricted to Selection: In the next step, the best sets of parameters are chosen according to their fitness value. Since our population is relatively small, we set the elite count variable to one, which means that only the best result is kept unaltered for the next generation. The rest of the chromosomes were chosen according to the selection function for the reproduction of a new generation. 4. Breed: In this step, a new population is created, either through crossover, which means a combination of two existing chromosomes, or by mutation. The crossover fraction defines the fraction of sets created by crossover. 5. Evaluation: The population of this new generation was again evaluated by calculating its fitness. The process is illustrated in figure Implementation in MATLAB We used the genetic algorithm function ga from the MATLAB Global Optimization Toolbox to implement the genetic algorithm for our purpose. The ga function calls the function which has to be optimized and hands over the angles: [parameters, Fval, exitflag, population] = ga(@tracked SO, 16, [], [],... [], [], lb, ub, [], options) The specified outputs were the final angles, the final value of the fitness function, the exit flag and the whole population of the last iteration. The inputs were the function tracked SO, the number of variables to be optimized, the lower and the upper bound of the angles and the options for the algorithm. The inequality constraints were defined in the function tracked SO itself. In the function tracked SO the SimMechanics model was simulated with the parameters given by the ga function. To do so, first all the parameters needed for the simulation had to initialized, including the values for the angles: function [F] = tracked SO(parameters)...

41 The Genetic algorithm Alpha 0 = parameters(1); Alpha 1 = parameters(2);... Alpha 15 = parameters(16); The names Alpha 0 to Alpha 15 are the ones recognized by the SimMechanics model. The simulation time had to be specified and the simulation had to be run. The model was called within a Try-Catch statement. This allowed us to override the default error behavior for the case that the simulation crashes. Without this statement, every time the simulation crashes the optimization would be aborted. try sim('model3d Final speed ctrl fine.mdl', 'SrcWorkspace', 'current',... 'StopTime','SimTime'); % Load data from Simulation Trackable1 = load('trackable1.mat'); Trackable2 = load('trackable2.mat'); % Inequality constraints if 0.1 > sum(parameters(2:16)) sum(parameters(2:16)) > 1.2*pi() % Punishment F = a + 2; else F = a Trackable1.ans(2,end) end catch % Punishment F = a + 2; end After simulating the model in the source workspace the results from the routine are loaded and the inequality constraints are checked. If they are not fulfilled the fitness function gets punished by setting it to f(α) = a + 2. If the constraints are satisfied the fitness function is calculated based on the results from the simulation. For the case that the simulation crashes ( catch statement) the fitness function gets punished as well. The initial values and the options were defined in the variable option which was created with the gaoptimset command: options = gaoptimset('initialpopulation', init parameters, 'Display',... 'diagnose', 'Generations', 40, 'EliteCount', 1, 'PlotInterval', 1,... 'CrossoverFraction', 0.5, 'UseParallel', 'always',... 'PopulationSize', 12,... A brief discussion of the options defined above - based on the guidelines of [13] - is given below: InitialPopulation: The initial population is given by the vector init parameters and a default variation of this vector. By giving initial parameters we could make sure, that at least one population of the first generation delivers a result. Since in the beginning the deviation in the mutation is quite high, good initial parameters were crucial. Display: Since the optimization is a very time consuming process, the display options were set to diagnose, which allowed us to get some insights after each generation of the optimization.

42 Chapter 5. The Optimization 32 Generations: The number of generation is a very important number, since it affects the quality of the results considerably. 40 generations is a trade off between computational effort and quality of the results. EliteCount: Since many shapes of the curved beam created by mutation don t work properly, it is important to stick to at least one functioning version after each generation. This is ensured by setting the EliteCount to 1. A number greater than one can lead to inbreeding and as a result of that to a homogeneous generation. CrossoverFraction: The crossover fraction was set to 0.5, which means that half of the new population was created by crossover. On the one hand, populations created by crossover are more likely to succeed, on the other hand if there re only populations created by crossover, the development goes strictly into one direction and it is likely that the global optimum is missed. The crossover fraction was set to 0.5 since this is a good compromise between the two arguments. MutationFcn: The mutation function was chosen to be mutationadaptfeasible since this is the only one that sticks strictly to the boundaries given. The mutation is executed based on a standard Gaussian distribution function with a standard deviation that decreases over the generations. That means that a random number taken from a Gaussian distribution with mean 0 was added to each entry of the parent vector. The standard deviation of this distribution was decreased with every iteration, so that the design space quasi narrowed. SelectionFcn: Our selection function is set to selectionroulette. The probability of a chromosome to be chosen is inversely proportional to its fitness value, that means the better the fitness, the higher the change to be chosen for reproduction. CrossoverFcn: The crossover function defines how the generations produced through crossover are build. Crossoverscattered is the default function. It creates a random binary vector with the length of the population. If the entry is 1 the chromosome from the first predecessor vector is chosen, if the value is zero, the value from the second one is chosen. PopulationSize: The population size defines how many individuals there are in one generation. Again, the lower the number the higher are the chances to find only a local minimum. On the other hand the higher the number the higher the computational effort. The size was set to 12. FitnessScalingFcn: With the fitscalingshiftlinear the raw fitness scores are converted in a range that is suitable for the selection function. The entire MATLAB function to optimize the model can be found in the Appendix E. 5.3 Experiments To check whether the optimization delivers satisfying results a couple of experiments were performed:

43 Problems during the implementation of the optimization Methodology to show the increase of the performance As a first step the increase of the performance in the simulation was looked at. To do so, three aspects were considered: As a first setting, the enhancement of the performance in the simulation was looked at in general. To do so the model was simulated twice, once before the optimization once afterwards and the hopping distances were compared to show that the optimization really works. The second part was set up to show that the optimization works for various motor speeds. The optimization was done with the same initial condition and the same optimization settings, but at different actuation speeds. In a third step, the influence of the optimization on the shape of the curved beam was looked at. The results from the second experiment were used to compare the shapes that resulted from different motor speeds to draw further conclusion on the design of the robot and the representation of the curved beam. Furthermore an analysis of the resonance frequencies of the different shapes after the optimization was done Methodology to show the repeatability of the results In the second experiment the repeatability of the optimization was investigated to show that the results are reliable and reproducible. The optimization was performed 7 times with the same initial conditions, same optimization options and the same actuation speed Methodology to validate the optimization results In the last and most important experiment, the optimization results were validated with the real world. To do so, first the performance of the initial robot was looked at in the real world and in the simulation at two different rotational speeds. The behavior found was stored as basis for the improvement. Based on the initial shape two optimizations were performed - one with each rotational speed - resulting in two different shapes. Each one of these shapes should be optimized exclusively for one actuation speed. The shapes found were recreated in the real world and the performance was looked at again. Both shapes were investigated with both actuation speeds to show that the optimization is depending on the rotational speed. The increase in performance was compared with the one from the simulation to make a meaningful statement about the optimization process developed in this work. 5.4 Problems during the implementation of the optimization During the implementation of this optimization process several problems occurred in the real world, as well as in the simulation Simulation problems In the simulation four major problems occurred which are discussed very briefly below:

44 Chapter 5. The Optimization 34 Figure 5.2: Two examples for a weirdly shaped robot. During the optimization process very often completely weird shapes appeared. Even though the performance of these shapes was often good, they had to be discarded due to the impossibility of a recreation in the real world. An example of such a shape is shown in figure 5.2. With the help of the inequality constraint this problem was reduced but not eliminated completely. A second problem was that the simulation crashed from time to time, which caused the whole optimization process to abort. Since the optimization process could take - depending on the settings - up to 1.5 days to complete this behavior was very annoying. To avoid this problem, the simulation was called from within a Try-Catch statement as mentioned above. Another problem was the falling over of the robot in the simulation. Initially it was possible that the robot flipped upside down without the simulation to end. Under certain circumstances it was even possible that the robots performance when flipped upside down was that good, that the genetic algorithm encouraged this behavior. An additional constraint was implemented in the model to stop immediately as soon as the vertical position of the point T lays under a certain threshold. The last problem deals with the high computational effort of the optimization. Even though MATLAB allows parallel computing when solving optimization problems, in our scenario this was not possible. The calculation of the fitness function is based on the results written out by the model. When the model is run more than once at the same time, these output files get messed up and a calculation of the fitness function gets pointless. A solution to this problem could not be found, the optimization has to be run without parallel computing Real world problems The implementation of the optimized shape in the real world posed some problems as well. First of all the sculpturing of the beam was very difficult to make it match the optimization result. To make the shape as close as possible to the one calculated, the results were printed out and adapted step by step by hand.

45 Problems during the implementation of the optimization 20 Speed Voltage curve omega [rad/s] Voltage [V] Figure 5.3: Relation between the rotational speed and the voltage. Another problem was to set the right actuation speed in the experiments. Since we build our robot without an encoder for the DC-motor, we had to estimate the motor velocity based on the voltage of the power supply. With the help of the tracking system, beforehand a speed-voltage curve was done, which defined the speed as: ω = mu + b (5.7) Here U denotes the voltage in volt and m and b are calculated based on the speedvoltage curve which can be seen in figure 5.3. m = 4.4 and b = 2.9.

46 Chapter 6 Results The results presented in this chapter are structured according to the experiments introduced in section Increase of the performance in the simulation For the first experiment, the model was optimized at a rotational speed of 17 rad/s. The model had initial angles so that already a hopping behavior could be observed. The model was optimized with the settings discussed in the previous chapter. The results are presented in figure Hopping with initial shape x [m] 0.05 Hopping with optimized shape x [m] Figure 6.1: On the top: The hopping behavior of the robot before optimization. Below: The hopping behavior after the optimization. 36

47 Increase of the performance in the simulation Hopping distance [m] rad/s 17 rad/s 18 rad/s 19 rad/s 20 rad/s 21 rad/s Optimizaion Loop versus max. hopping distance Iteration Hopping distance [m] rad/s 17 rad/s 18 rad/s 19 rad/s 20 rad/s 21 rad/s Optimizaion Loop versus average hopping distance Iteration Figure 6.2: On the top: The maximal hopping distance in each generation at different actuation speeds. Below: The average hopping distance of each generation at different actuation speeds. The next optimization was started at the same initial angles, but with different actuation speeds. In figure 6.2 it can be seen that regardless of the rotational speed a great improvement of the hopping distance can be seen. On the upper graph the maximal hopping distance in each generation is shown, in the lower graph the average hopping distance of the whole generation is shown. The different shapes resulting from this optimization can be seen in figure 6.3. As can be seen the model looks very similar, but still slight differences can be spotted. To quantify the differences between the different results, the shapes were analyzed according to their shape parameters L 0, d h and e. Further the longitudinal res- Figure 6.3: The different shapes of the curved beam after the optimizations. From left to right optimized at 17, 18, 19, 20 and 21 rad/s.

48 Chapter 6. Results 38 onance frequency ζ L of each shape was measured in the simulation, to show that the optimization also has an impact on the free vibration modes of the curved beam. ω [rad/s] L 0 [cm] d h [cm] e ζ L [/s] Repeatability of the optimization results An important aspect of this optimization approach is the repeatability. A reliable optimization must deliver results that can be reproduced in a reasonable quality. To investigate the repeatability, the model was optimized 7 times at the same rotational speed, the same settings and the same initial parameters. The results can be seen in figure Optimizaion Loop versus max. hopping distance Hopping distance [m] Iteration Figure 6.4: Maximal hopping distance in simulation actuated at 17 rad/s for various optimizations.

49 Validation of the optimization in the real world Simulation of initial model at 17 rad/s Real world experiment of initial model at 17 rad/s z [m] 0.02 z [m] x [m] Simulation of optimized model at 17 rad/s x [m] Real world experiment of optimized model at 17 rad/s z [m] 0.02 z [m] x [m] Simulation of optimized model at 19 rad/s x [m] Real world experiment of optimized model at 19 rad/s z [m] 0.02 z [m] x [m] x [m] Figure 6.5: Comparison of the simulation results and the results from the real world at 17 rad/s. 6.3 Validation of the optimization in the real world To validate the optimized results from the simulation in the real world, three different experiments were performed in the real world and in the simulation: 1. The robot was simulated and tracked with a rotational speed of 17 rad/s. 2. The robot was optimized at a rotational speed of 17 rad/s. After that it was again simulated, respectively tracked at a rotational speed of 17 rad/s. 3. Finally the robot which was optimized at a rotational speed of 17 rad/s was also simulated/tracked at a rotational speed of 19 rad/s, to show that the optimization is explicitly for one rotational speed. The results can be seen in figure 6.5. On the left side the results from the simulation are shown, on the right side the results from the real world robot can be seen. The same experiment was performed a second time. The only difference was that the model was optimized at a rotational speed of 19 rad/s. The results can be seen in figure 6.6. The results from the figures 6.5 and 6.6 are presented again in figure 6.7. In this plot only the performance of the real world robot actuated at 17 and 19 rad/s are shown. The comparison is done for robot with the initial shape of the curved beam, the robot optimized at an actuation speed of 17 rad/s and the robot optimized at 19 rad/s.

50 Chapter 6. Results 40 Simulation of initial model at 19 rad/s Real world experiment of initial model at 19 rad/s z [m] 0.02 z [m] x [m] Simulation of optimized model at 19 rad/s x [m] Real world experiment of optimized model at 17 rad/s z [m] 0.02 z [m] x [m] Simulation of optimized model at 19 rad/s x [m] Real world experiment of optimized model at 19 rad/s z [m] 0.02 z [m] x [m] x [m] Figure 6.6: Comparison of the simulation results and the results from the real world at 19 rad/s rad/s 19 rad/s 1 Hopping distance [m] Initial robot Robot optimized at 17 rad/s Robot optimized at 19 rad/s Figure 6.7: The performance of the real world hopping robot actuated at 17 and 19 rad/s. From left to right the initial shape of the curved beam, the shape optimized at 17 rad/s and the shape optimized at 19 rad/s are used.

51 Chapter 7 Analysis of the results 7.1 Analysis of the increase of the performance in the simulation As can be seen in figures 6.1 and 6.2, the performance was increased significantly. Depending on the actuation speed the hopping distance was increased by almost 600 %. This great improvement could not have been ecspected and shows that the optimization and also the representation of the robot in the simulation were chosen appropriate. It can be seen as well that the optimization process works at different actuation speeds and with different initial models. In the figures 6.2 and 6.3 it can also be observed that the performance improves over the whole optimization process. Also in the last generations a slight change to the better can be stated. This can be explained by the decreasing of the variance of the mutation function: In the last generations the shapes only experience minor changes. The fact that the the improvement is distributed over the whole optimization process is an indicator that the right parameters were chosen. It can be seen that the performance of the robot is increased no matter how good the performance was initially. In figure 6.2 it can be seen that the initial hopping distance of the model actuated at 18 and 19 rad/s is already pretty descent. Nevertheless, the performance is increased even more. The shape created by the optimization look pretty much like they were expected before hand. It has to be said that, even though a slight tendency can be seen that a higher actuation speeds induces smaller value for e, no clear correlation between the resonance frequencies and these parameters can be seen. A generalized formula, on how to design the curved beam to achieve which behavior, could not be found. This is a good reason why an automated approach for the design - as presented in this work - is useful. Further it can be seen from the results, that the longitudinal frequencies of the optimized models were shifted towards the actuation speed. This confirms the thesis that the models perform the best when the actuation speed and the resonance frequency match. 41

52 Chapter 7. Analysis of the results Analysis of the results of the repeatability In figure 6.3 the repeatability of the optimization was investigated. The results showed that every time the optimization was performed, at significant change to the better could be stated. As can be seen, the performance was always increased by at least 200 %. The maximum hopping distance lays between 0.97 and 1.48 m, only one optimization resulted in a hopping distance of 2.05 m which lays outside of the standard deviation. These results show us, that on the one hand, the optimization is reproducible up to a certain degree and that they are reliable to happen. On the other hand, the fact that one optimization delivered a notably better performance than all the others shows us, that the global maximum is not found with this optimization. This is not a surprising insight, since a genetic algorithm is never capable of doing so. Nevertheless, the average result could still lay closer to the maximum. There is still room for improvement. 7.3 Analysis of the optimization results in the real world As can be seen in the figures 6.5 and 6.6 the performance was improved significantly in the real world. The absolute improvement was even stronger in the real world than in the simulation. It can be seen that the optimization worked properly in the real world for both actuation speeds. In figure 6.5 it can be seen, that the model which was optimized at an actuation speed of 17 rad/s, performs a lot better when actuated at 17 rad/s. When actuated at 19 rad/s, the improvement is less strongly. This confirms again the assumption that the optimization delivers different shapes depending on the actuation speed. The same behavior is also shown in figure 6.7, where the performance gained from the experiments conducted in the real world are summarized.

53 Chapter 8 Conclusion The indisputable improvement of the performance of the real world robot states clearly that the design based on a optimization approach works. Even though slight weaknesses can be spotted in the repeatability and the quantitative accuracy of the results, the fact that it is possible to design the beam based on this approach leads to a positive conclusion. Following points should be highlighted: The representation of the Curved Beam Hopping robot as a series of rigid bodies connected with spring damper elements works nicely. The results are qualitatively very good and the computational effort is acceptable. Quantitatively the results from the simulation might be improved. The Ground Contact model as used in this simulation allows a realistic representation of the reality. The optimization process based on a genetic algorithm works nicely in the simulation and delivers reliable results. The improvement of the performance is solid, even though it is only repeatable to a certain degree. The design parameters found by the optimization worked perfectly in the real world; a great improvement of the performance could be stated in the real world experiments. 43

54 Chapter 9 Future Work Even though the final statement of this work is positive, there are still several open questions and challenges to tackle: The repeatability of the optimizations could still be improved, To do so different strategies could be chosen, either by finding better parameters of the already existing algorithm, or by implementing a new approach all together. The optimization of the robot was limited to the design of the curved beam. Future projects could also include additional factors into the optimization, such as rotating speed, rotating mass, etc. This could be done with reasonable effort. The optimizations performed in this work were always focused on the increase of the hopping speed. Nevertheless, other optimization goals - such as energy efficiency or maximum hopping height - might be interesting as well. The quantitative evidence of the model used in the simulation could still be improved. This could be done by finding better parameters, adjusting the model (e.g. by increasing the number of increments in the beam) or by a completely new approach. 44

55 Appendix A The SimMechanics Simulation A.1 The model hierarchy An overview of the model is presented in figure A.1. The model has the following hierarchical structure: model3d Final speed ctrl fine Actuation Animation Curved Beam Increments 0 to 15 Ground Interaction Model F and R X-Axis Stiction-Friction Model 45

56 Appendix A. The SimMechanics Simulation 46 A.2 Model overview An overview of the SimMechanics model is presented in figure A.1. The blue subsystem includes the actuator block, the pink subsystem the animation and the green subsystem contains the GCM. In the red blocks the foot and the curved beam are illustrated. The subsystems are connected by the SimMechanics signal that they share. In figure A.1 also the blocks that define the simulation environment - such as the Ground and the Machine environment block - can be seen. Figure A.1: Overview of the SimMechanics model.

57 47 A.3. Actuation A.3 Actuation In figure A.2 the subsystem for the actuation of the model is shown. It can be seen that the rotating mass is modeled with to connected rigid bodies (one for the radius and another one for the rotating mass itself) that rotate around a revolute joint. This joint is actuated with the motor subsystem, where a simple feed-forward speed control is implemented. The GoTo blocks in the subsystem are used for the animation of the model. Figure A.2: Speed control of the rotating mass.

58 Appendix A. The SimMechanics Simulation 48 A.4 Animation The main part of the animation is the VR-Sink block, that connects the Sim- Mechanics model with the VRML world. The world is defined in a separate file. The variables from the simulation are handed over to the animation subsystem via GoTo respectively From blocks. Figure A.3: Integration of the VR-Sink into the model.

59 49 A.5. Curved Beam A.5 Curved Beam The curved beam is represented with a series of identical subsystems, as can be seen in figure A.4. The SimMechanics signal is passed from one subsystem to the other. Figure A.4: The curved beam in the model.

60 Appendix A. The SimMechanics Simulation 50 Each increment of the curved beam consists of a rigid body, a revolute joint and a spring damper system. In addition each increment includes a body sensor and a GoTo block, to hand over the coordinates and the orientation of the increments to the animation subsystem. Figure A.5: One increment of the curved beam.

61 51 A.6. Ground Contact model A.6 Ground Contact model The subsystem for the GCM. The forces in y- and z-direction are calculated directly, the ground reaction forces in x-direction are calculated in the block subsystem 2. The subsystems are only active if the condition for Stance is fulfilled. This is the case when the vertical position of the foot point is below zero. Figure A.6: GCM of the front right foot.

62 Appendix A. The SimMechanics Simulation 52 The ground reaction forces in x-direction are shown in figure A.7. The JointStictionActuator blocks calculates the reaction forces in x-direction and switches between friction force and stiction. The inputs of this block come from body sensor blocks and predefined constants. Figure A.7: x-axis Stiction-Friction model.