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The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015 DOI Number: 10.6567/IFToMM.14TH.WC.OS13.001 Optimal Design of Klann-based walking Mechanism for Water-running Robots HyunGyu Kim, JaeNeung Choi, TaeWon Seo School of Mechanical Engineering, Yeungnam University, Gyeongsan, 712-749, Korea Correspondence e-mail to: taewon_seoyu.ac.kr Kyungmin Jeong Nuclear Convergence Technology Division, Korea Atomic Energy Research Institute, Daejeon, 305-353, Korea Abstract: Recently, many studies have been conducted to develop robots inspired by living creatures, but these robots have limitations when operating in various environments. In previous research, we developed an amphibious robot that could operate on a water surface and on the ground. The robot has good stability, but its running speed is not sufficient. In this study, we optimized the Klann leg mechanism of the robot to increase the running speed based on water surface operation. First, we modeled the interaction force between the water and the feet of the robot. We also numerically calculated the running speed when operating on water as the object function of the optimization process. We optimized the running speed using level average analysis. The result of optimization was used for ground locomotion, and the optimized running speeds were compared with the results of previous research. The optimized Klann mechanism will be used as the leg mechanism in an amphibious robot for running on water or on the ground. Keywords: Amphibious robot, Klann mechanism, optimal design, level average analysis 1 Introduction Research inspired by the characteristics of living things is very popular for developing new robotic platforms. Living things have evolved for a long time to survive in many environments. For this reason, they have abilities which humans lack. For example, a tank-like module-based climbing robot was developed by the Nanorobotics Laboratory at Carnegie Mellon University. This robot could climb walls using the Van-der-Waals force between the robots feet and the wall 1. This robot was inspired by gecko lizards. A robot inspired by a cheetah was developed by Kim et al. 2, which could run at up to 33 mph and jump up to 0.3 m. Linkage structures are very useful to realize the trajectories of living things. Linkage structures have advantages because they can generate various kinds of trajectories using only one actuator. So, researchers have made light robots by using linkage structures. DASH can operate on the ground using a linkage structure and one motor 3, 4. The body of DASH consists of an inner part and external part connected with a linkage structure. The movement of these two parts makes the trajectory for each of six legs. The linkage structure also reduced the robot mass by using one motor and because the material of the robot is very flexible. In the case of flying robots, mass is a very important factor. So, many researchers have used linkage structures to realize bird-flapping locomotion 5. In our previous research, we designed an amphibious robot platform inspired by a basilisk lizard. Basilisk lizards are well known because they can run on a water surface. They can also run on the ground. They use the drag force of water to generate velocity and a specific trajectory of their feet 6-9. Many researchers have been interested in the locomotion of basilisk lizards. The first robot inspired by the basilisk lizard was proposed by Sitti et al. 10. They compared the performance of different foot designs 11. We also conducted research related to water-running robots 12. The purpose of a previous study was designing a robot that can operate on a water surface and on the ground 13. We used the buoyancy generated by Styrofoam to float on the water surface. Using the Klann mechanism, the robot could generate drag force in the water and run on the ground. The Klann mechanism is appropriate for amphibious locomotion for robots that can float on water, because the mechanism was developed for walking devices. The robot and Klann mechanism are introduced in Section 2. We performed a numerical analysis and experiment to confirm the running speed and stability in the two environments examined in the previous research. These were conducted at different frequencies of the motor ranging from 1.7Hz to 2.5Hz. The robot shows stable locomotion. Roll and yaw motions are not generated, because the structure of the robot eliminates the moments in these directions. So, we consider only pitch motion, which was smaller than 10 degrees in all cases. The running speed is increased as the frequency of the motor is increased. However, the speed was not high enough for free operation. In this study, we optimized the length of links to increase the running speed. First of all, we numerically analyzed the position and velocity of the Klann linkage. Then, we defined variables, an object function, and constraints. Using level average analysis, we obtained optimized variables through iterative calculation. This paper is organized as follows. Section 2 introduces the specifications of the robot platform and the analysis process of the Klann mechanism. Section 3 defines the interaction force between the water surface and feet of the robot. Then, locomotion for ground operation is defined. Section 4 presents the optimization process, variables, object function, and constraints. Then, the results of optimization are presented. Finally, in Section 5, a conclusion is given. 2 Robot prototype In this Section, robot specifications such as the length, mass, and moment of inertia are presented. Then, the Klann mechanism is introduced and analyzed. The lengths of the links are the variables of optimization. 2.1 Robot specifications Fig. 1A shows the robot platform. This robot has two DC (Direct Current) motors connected in parallel with a middle driven shaft. The shaft is connected with front and back driven shafts using timing belts and pulleys. So, all legs are dependent on two motors. We used spherical Styrofoam as the feet. The robot can float on water using the Styrofoam feet, which generate buoyancy. Then, using the trajectory and velocity of the Klann mechanism, the feet generate drag force in the water for moving in the forward direction. Two legs connected with one driven shaft have a phase difference of 180 degrees. Therefore, three legs which have the same phase and form a triangle move together. The robot uses a tripod gait like in Fig. 1B. We designed the robot to eliminate moments in the roll and yaw directions by having the center of the triangle pass through the center of mass of the robot. However, as the feet rotate, the center points of forces form a line which passes through the center of mass, like the yellow line in Fig. 1B. Therefore, pitch motion could not be eliminated structurally. For this reason, we considered pitch motion in the analysis of the robot locomotion. This part of the analysis is presented in Section 3. The specifications of the robot platform are indicated in Table 1. 2.2 Klann mechanism The first Klann mechanism was designed for walking locomotion. The mechanism consists of six links and has only one driving link connected with a motor. So, as the motor is operated, the end effector of the Klann mechanis Table 1 Specifications of the robot platform SpecificationsValues Length408 mm Width230 mm Mass208.6 mm Radius of the feet 30 mm Moment of inertia (roll) 0.001 kg.m2 Moment of inertia (pitch)0.008 kg.m2 Moment of inertia (yaw) 0.009 kg.m2 -m generates specific trajectories and velocities. Fig. 2 shows a schematic view of the Klann mechanism. To analyze the Klann mechanism, we changed links into vectors like in Fig. 2. There are ten vectors and three angles, which can be variables. However, we used only ten variables in this optimization: seven vectors and three angles. Three vectors forming a triangle such as l4, l8, and l9 were not considered in the optimization. To calculate the position and velocity of the end effector, we used an Euler equation (1). An Euler equation is convenient for summing vectors. Then, by summing vectors which connect with the input point of l1 and end effector of lh, we calculated the position and velocity. Equation (2) describes the position and velocity of the end effector, which are used to calculate the running speed and pitch motion in two environments in the analysis. nn i ie n sin+cos= (1) where n is the angle between the horizontal line and each link. n is the number of vectors, which ranges from 1 to 10. i is an imaginary number. n n i nnn i nn ewilVelocity elPosition = ,= (2) where ln is the length of the each link, and wn is the angular velocity of each link. 3 Analysis In this Section, the running speed and pitch motion in two environments are analyzed. The running speeds when the d c a e b Front Middle Back Body A B Figure 1. A. 3-D modeling of the robot prototype. a) DC motors, b) Timing belts, c). Klann mechanism, d) Spherical Styrofoam feet, e) Gyro sensor to measure pitching angle. B. Schematic top view of the robot. Yellow solid line indicates the center of forces. Green point indicates the center of mass of the robot platform. Orange arrows denote the moving direction. l8 l4 l3 l2 l5 l6 l9 l1 l7 lh Figure 2. Schematic view of the Klann mechanism. The solid arrows indicate the vector of the links. The orange curved arrow indicates rotation direction of the motor. The orange arrow indicates the moving direction of the robot. The red dashed line is the trajectory of the Klann linkage. robot runs on water are the object function for optimization. 3.1 Interaction force between water and feet In this section, we define the interaction force between the water and the robot feet. There are some conditions in the analysis of the interaction force that depend on the moving direction of the feet. In the case of the horizontal direction, when a foot moves in the direction opposite to that of the robot, the reaction force of the drag force generates a positive force in the forward moving direction. However, a negative force is generated in the opposite condition. These two conditions are applied to the vertical direction. When a foot moves in the downward direction, the reaction forces of the drag force generate a positive force in the upward direction. A negative force is also generated in the opposite condition. Due to these four conditions, the submerged depth is continually changed. The depth is an important variable because the buoyancy and drag force of the water change with the depth. To define the drag force of the water, we referred to a study conducted by Glasheen and McMahon 8. They defined the modeling for drag force for a circular plate foot. In the present study, the feet have a spherical shape. So, we assumed that the projection of the submerged areas of spherical feet is the same as that of a circular plate. In addition, we did not consider the vibration of the water surface. A schematic view of the interaction force is indicated in Fig. 3. Equation (3) describes the drag force of the water: )()(+)()(5 . 0(=)( 2* tStghtStuCtD Df (3) where Df (t) is the drag force between the water and the feet. CD* is the coefficient of water drag and is equal to 0.707. is the density of the water. u(t) is the velocity of the foot. h(t) is the submerged depth, which is changed according to the buoyancy, drag force of water, and pitch motion. S(t) is the submerged area, which was defined as passing through the center of the foot and is perpendicular to the rotation axis of a foot. The submerged area is determined by the submerged depth. Buoyancy is generated when the feet are submerged under the water surface, because the feet material is Styrofoam, as shown in Fig. 3. The buoyancy influences the two aforementioned conditions associated with the vertical direction, because the submerged depth determining the buoyancy is changed by movement of the feet. As a result, buoyancy also influences the submerged depth, and we considered it in the analysis. Buoyancy is described as follows: x 0 2 22 n ,B dyryr)t (F (4-A) 0th r2th r2th0 0 r2 th x (4-B) where FB is the buoyancy, is the density of the water, r is the radius of the feet equal to 30 mm, n is the number of submerged feet, and x is the submerged depth. The submerged depth is divided into three cases, like in equation (4-B). If the submerged depth is larger than the diameter of a foot, it means that a foot is totally submerged and the buoyancy is maximum. If the submerged depth is smaller than zero, it means a foot is not submerged and there is no buoyancy. Finally, if the submerged area is between the diameter of a foot and zero, the submerged depth is continually changing. The pitching angle also influences the submerged depth. The pitching angle is generated by drag force in the vertical direction and buoyancy, as in Fig. 4. When the drag force in the vertical direction and buoyancy are generated at the foot, torque is generated because of the distance between these forces and the center of mass. The generated pitching angles change the submerged depth of each foot. Then, the pitching angles generate additional buoyancy, like in Fig. 4. As a result, forces including the drag force in the vertical direction, buoyancy, and pitching angles influence the submerged depth. Since the submerged depth is related to the drag force in the horizontal direction, it is a very important variable for calculating the running speed and pitching angle. Equation (5) describes the condition to calculate the submerged area: - n D massnB g ttF RtF 0 , )(sin()( )( (5) Water h FD FD FD, y FD, x FD, y FD, x y x FB FB Foot Figure 3. Schematic view of the interaction force between the water and the feet. The foot consists of spherical Styrofoam. h is the submerged depth. FD is the drag force of the water. FD, x and FD, y are components of the drag force (FD) in the x and y directions. FB is the buoyancy due to the submerged depth. Red lines indicate the submerged projection area. Orange curved arrows indicate the rotation direction. hp Water p Additional buoyancy Figure 4. Schematic view of the pitch motion. p is the generated angle due to the pitch motion. hp is the additional distant due to the pitch motion. The orange curved arrow indicates the rotation direction due to the pitch motion where n is the number of submerged feet, Rmass is the total mass of the robot, t is the angle between the water surface and moving direction of the feet, and g is the acceleration of gravity equal to 9.8 m/s. 3.2 Ground locomotion In ground operation, we assumed that there is no slip between the feet and the ground. The velocity of the feet can be considered as the running speed of the robot. We think that it is difficult for pitching angle to occur, because the robot uses a tripod gait, and the center of forces passes through the center of mass symmetrically in the case of pitch motion in ground operation. Three feet make contact with the ground while forming a triangle. So, if the center of mass is in the triangle, there can be no pitching angle. 4 Optimization In this section, the optimization process is presented. We used the level average analysis method for optimization. In addition, variables for optimization were defined as ten link lengths, which are defined in Section 2. The object function for optimization is the running speed when the robot operates on the water surface. 4.1 Level average analysis To do the optimization, we first defined an orthogonal array to reduce the number of iterations. Because there are ten variables, we used L32 (2149), which has one variable of two levels and nine variables of four levels. In addition, this orthogonal array has thirty-two different cases. The differences in level of all ten variables are 0.05 mm. This difference in levels is decreased for each iteration by 0.001 mm. We calculated thirty-two signal-to-noise ratios (S/N ratio) for each case for each iteration. Then, we chose ten new variables for the next iteration through sensitivity analysis. The ten variables are optimized by these repeated processes. Equation (6) describes the S/N ratio. - 2 i Y 1 n 1 log10ratioN/S (6) where n is the number of environments, and Yi is the value of the object function. Increasing the object function (Yi) means that the running speed is improved, because the S/N ratio is a negative value. It is possible to analyze the sensitivity by averaging the S/N ratios of the same level for each variable. In this sensitivity analysis, the maximum values for each variable become new variables for the next iteration. As the iterations are increased, the variables reach the optimal values. Also, the S/N ratios are increased. We also defined constraints to the size and mass of the robot platform. If we do not define the constraints, the variables will grow to infinity, because the running speed is increased as the distance between the end effector and input point is increased. The constraint is that the trajectory of the Klann linkage cannot exceed the input point. The algorithm for the optimization process is given in Fig. 5. 4.2 Results of the optimization The optimization gave the optimized variables after thirty- four iterations. Fig. 6 shows the object function according to the iterations. As the iteration number is increased, the running speed on the water is also increased. The RMS value of the running speed is increased from 0.425 m/s to 0.55 m/s. After thirty-four iterations, the x coordinate of the optimized trajectory exceeded the input point. The initial and optimized variables are indicated in Table 2. The trajectories before and after optimization are indicated in Fig. 7. There are some distinct features in the optimized trajectory compared wi