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AnExperimentalStudyofThree-Dimensional PassiveDynamicWalkingwithFlatFeetandAnkleSprings 141 Fig. 15. Counting steps in passive walking. The first landing is counted as the zeroth step and the second landing as the first. Successful Steps Number of times (in 100 launch) 3920 N/m 6180 N/m 9320 N/m 13530 N/m ∞ N/m 0 40 12 7 12 40 1 51 68 68 70 49 2 9 14 19 12 10 3 0 3 6 5 1 4 0 3 0 1 0 Expectation 0.78 1.17 1.24 1.13 0.72 Table 3. Experimental result for each pitch angle spring constant Fig. 16. Comparison of the foot motion for varying spring stiffness motions are coordinated. From (Narukawa et al., 2009a), the value of the spring constant of the extension spring for roll motion is determined to be 4900 N/m with a torsional spring stiffness of about 9 N-m/rad. Figure 14 shows the relationship between the ankle angle and the torque applied by the springs for pitch motion when the spring constant is 3920, 6180, 9320, and 13530 N/m. Fig. 17. Rebound at foot impact (3920 N/m) 5. Experimental Results Experimental conditions were as follows. The slope angle was about 1.6 degrees; the slope was about 3.6 m long and 0.6 m wide. In each trial, the walker was started by hand from the top of the slope. 5.1 Successful Steps at Different Spring Constants for Pitch Motion Figure 15 shows how we count the number of realized steps. After the launch in Fig. 15 (a), first landing is counted as the zeroth step and second landing is the first step, as shown in Fig. 15 (b) and (c). When the swing leg lands and the heel is behind the toe of the stance foot, we do not count the landing as a successful step and regard the walking as a failure, as shown in Fig. 15 (e) and (f). In some situations, we do not count a landing as a successful step even though the swing leg lands in front of the stance leg. Table 3 shows the experimen- tal results. We changed pitch spring constant of the ankles and launched the biped walker 100 times for each settings. The spring constant ∞ [N/m] means that we locked the ankle pitch movements with wires instead of springs. However, the feet are not ideal rigid bodies because we attached sponge sheets to the soles of the feet. The result shows that a medium spring constant effectively stabilizes passive walking. High and low spring constants are in- effective because unsuccessful trials (0 step) occurred frequently, producing a low expectation of successful steps. 5.2 Foot Motion with the Ground The spring stiffness of the pitch motion affects the foot motion with the floor. Figure 16 com- pares the foot motion for different spring stiffness. When the spring constant is 3920 N/m, which is equal to a torsional spring stiffness of about 7 Nm/rad, the foot of the stance leg remains in full contact with the floor until the heel of the swing leg touches the floor. Next, the front foot fully impacts the floor and rebound occurs, as indicated in Fig. 17. On the other hand, when the torsional spring stiffness is large, e.g., the spring constant is 9320 N/m, the rotation of the stance foot around the toe occurs before the swing foot touches the floor and the rebound after the full contact of the front foot is dramatically reduced. A high torsional spring stiffness for pitch motion leads to a smooth transition at the exchange of the stance leg. When the torsional spring stiffness is very large, the pitch angle is always 0, the stance foot almost always rotates around the heel or toe, and rebound does not occur. 6. Conclusions This paper presents a simple 3D passive biped walker with flat feet and ankle springs. Experi- mental tests were performed to investigate the effects of torsional spring stiffness on the pitch CuttingEdgeRobotics2010142 motion at the ankle joints of the walker. When the spring stiffness is low, oscillating motion is induced by the impact of the feet with the ground. Experimental results showed that using springs with appropriate torsional spring stiffness effectively reduces the oscillating motion. The rebound of the front foot after full contact with the ground reduces dramatically with appropriate torsional spring stiffness. Appropriate stiffness enables the biped walker to walk smoothly and also stabilizes the walker. However, when the spring stiffness is either high or low, it become difficult for the walker to walk. ACKNOWLEDGMENTS This work was supported in part by Grant in Aid for the Global Center of Excellence Program for “Center for Education and Research of Symbiotic, Safe and Secure System Design" from the Ministry of Education, Culture, Sports, Science and Technology in Japan. 7. References Adolfsson, J., Dankowicz, H., and Nordmark, A. (2001). 3D passive walkers: Finding periodic gaits in the presence of discontinuities. Nonlinear Dynamics, 24(2):205–229. Coleman, M. J., Garcia, M., Mombaur, K., and Ruina, A. (2001). Prediction of stable walking for a toy that cannot stand. Physical Review E, 64(2):22901–. Coleman, M. J. and Ruina, A. (1998). An uncontrolled walking toy that cannot stand still. Physical Review Letters, 80(16):3658–3661. Collins, S. and Ruina, A. (2005). A bipedal walking robot with efficient and human-like gait. Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pages 1983–1988. Collins, S., Ruina, A., Tedrake, R., and Wisse, M. (2005). Efficient bipedal robots based on passive-dynamic walkers. Science, 307(5712):1082–1085. Collins, S., Wisse, M., and Ruina, A. (2001). A three-dimensional passive-dynamic walking robot with two legs and knees. International Journal of Robotics Research, 20(7):607– 615. Garcia, M. (1999). Stability, scaling, and chaos in passive-dynamic gait models. PhD thesis, Cornell University. Garcia, M., Chatterjee, A., and Ruina, A. (2000). Efficiency, speed, and scaling of two- dimensional passive-dynamic walking. Dynamics and Stability of Systems, 15(2):75–99. Garcia, M., Chatterjee, A., Ruina, A., and Coleman, M. (1998). The simplest walking model: Stability, complexity, and scaling. Journal of Biomechanical Engineering-Transactions of the ASME, 120(2):281–288. Goswami, A., Espiau, B., and Keramane, A. (1996). Limit cycles and their stability in a pas- sive bipedal gait. Proceedings of the 1996 IEEE International Conference on Robotics and Automation, pages 246–251. Goswami, A., Thuilot, B., and Espiau, B. (1998). A study of the passive gait of a compass- like biped robot: Symmetry and chaos. International Journal of Robotics Research, 17(12):1282–1301. Hobbelen, D. G. E. and Wisse, M. (2007). Limit cycle walking. In Hackel, M., editor, Humanoid Robots: Human-like Machines, pages 277–294. I-Tech Education and Publishing, Vi- enna, Austria. Ikemata, Y., Sano, A., and Fujimoto, H. (2006). A physical principle of gait generation and its stabilization derived from mechanism of fixed point. Proceedings of the 2006 Conference on International Robotics and Automation, pages 836–841. Kinugasa, T., Kotake, K., Haruki, T., Tanaka, H., and Yoshida, K. (2008). 3D passive dynamic walker with sprung ankle and flat foot -a design method by natural frequency in- dex without yaw and roll compensator Proceedings of the 2008 JSME Conference on Robotics and Mechatronics, pages 1P1–B12–. Kuo, A. D. (1999). Stabilization of lateral motion in passive dynamic walking. International Journal of Robotics Research, 18(9):917–930. McGeer, T. (1990). Passive dynamic walking. International Journal of Robotics Research, 9(2):62– 82. McGeer, T. (1993). Passive dynamic biped catalogue, 1991. In Chatila, R. and Hirzinger, G., editors, Experimental Robotics II: The 2nd International Symposium, Toulouse, France, June 25-27 1991, pages 465–490. Springer-Verlag. McMahon, T. A. (1984). Mechanics of locomotion. International Journal of Robotics Research, 3(2):4–28. Narukawa, T., Yokoyama, K., Takahashi, M., and Yoshida, K. (2008). A simple 3D straight- legged passive walker with flat feet and ankle springs. IEEE/RSJ International Confer- ence on Intelligent Robots and Systems, pages 2952–2957. Narukawa, T., Yokoyama, K., Takahashi, M., and Yoshida, K. (2009a). Design and construction of a simple 3D straight-legged passive walker with flat feet and ankle springs. JSME Journal of System Design and Dynamics, 3(1):1–12. Narukawa, T., Takahashi, M., and Yoshida, K. (2009b). Design and Stability Analysis of a 3D Rimless Wheel with Flat Feet and Ankle Springs. JSME Journal of System Design and Dynamics, 3(3):258-269. Schwab, A. L. and Wisse, M. (2001). Basin of attraction of the simplest walking model. Pro- ceedings of the ASME Design Engineering Technical Conference, pages DETC2001/VIB– 21363–. Tedrake, R., Zhang, T. W., Fong, M F., and Seung, H. S. (2004). Actuating a simple 3D pas- sive dynamic walker. Proceedings of the IEEE International Conference on Robotics and Automation, 2004(5):4656–4661. Tedrake, R. L. (2004). Applied Optimal Control for Dynamically Stable Legged Locomotion. PhD thesis, Massachusetts institute of technology. Wisse, M., Hobbelen, D. G. E., Rotteveel, R. J. J., Anderson, S. O., and Zeglin, G. J. (2006). Ankle springs instead of arc-shaped feet for passive dynamic walkers. Proceedings of the IEEE-RAS International Conference on Humanoid Robots, pages 110–116. Wisse, M. and Schwab, A. L. (2005). Skateboards, bicycles, and three-dimensional biped walk- ing machines: Velocity-dependent stability by means of lean-to-yaw coupling. Inter- national Journal of Robotics Research, 24(6):417–429. Wisse, M., Schwab, A. L., van der Linde, R. Q., and van der Helm, F. C. T. (2005). How to keep from falling forward: Elementary swing leg action for passive dynamic walkers. IEEE Transactions on Robotics, 21(3):393–401. Wisse, M., Schwab, A. L., and vander Linde, R. Q. (2001). A 3D passive dynamic biped with yaw and roll compensation. Robotica, 19:275–284. Wisse, M. and van Frankenhuyzen, J. (2003). Design and construction of mike; a 2D au- tonomous biped based on passive dynamic walking. Proceedings of the Second In- ternational Symposium on Adaptive Motion of Animals and Machines, pages 4–8. AnExperimentalStudyofThree-Dimensional PassiveDynamicWalkingwithFlatFeetandAnkleSprings 143 motion at the ankle joints of the walker. When the spring stiffness is low, oscillating motion is induced by the impact of the feet with the ground. Experimental results showed that using springs with appropriate torsional spring stiffness effectively reduces the oscillating motion. The rebound of the front foot after full contact with the ground reduces dramatically with appropriate torsional spring stiffness. Appropriate stiffness enables the biped walker to walk smoothly and also stabilizes the walker. However, when the spring stiffness is either high or low, it become difficult for the walker to walk. ACKNOWLEDGMENTS This work was supported in part by Grant in Aid for the Global Center of Excellence Program for “Center for Education and Research of Symbiotic, Safe and Secure System Design" from the Ministry of Education, Culture, Sports, Science and Technology in Japan. 7. References Adolfsson, J., Dankowicz, H., and Nordmark, A. (2001). 3D passive walkers: Finding periodic gaits in the presence of discontinuities. Nonlinear Dynamics, 24(2):205–229. Coleman, M. J., Garcia, M., Mombaur, K., and Ruina, A. (2001). Prediction of stable walking for a toy that cannot stand. Physical Review E, 64(2):22901–. Coleman, M. J. and Ruina, A. (1998). An uncontrolled walking toy that cannot stand still. Physical Review Letters, 80(16):3658–3661. Collins, S. and Ruina, A. (2005). A bipedal walking robot with efficient and human-like gait. Proceedings of the 2005 IEEE International Conference on Robotics and Automation, pages 1983–1988. Collins, S., Ruina, A., Tedrake, R., and Wisse, M. (2005). Efficient bipedal robots based on passive-dynamic walkers. Science, 307(5712):1082–1085. Collins, S., Wisse, M., and Ruina, A. (2001). A three-dimensional passive-dynamic walking robot with two legs and knees. International Journal of Robotics Research, 20(7):607– 615. Garcia, M. (1999). Stability, scaling, and chaos in passive-dynamic gait models. PhD thesis, Cornell University. Garcia, M., Chatterjee, A., and Ruina, A. (2000). Efficiency, speed, and scaling of two- dimensional passive-dynamic walking. Dynamics and Stability of Systems, 15(2):75–99. Garcia, M., Chatterjee, A., Ruina, A., and Coleman, M. (1998). The simplest walking model: Stability, complexity, and scaling. Journal of Biomechanical Engineering-Transactions of the ASME, 120(2):281–288. Goswami, A., Espiau, B., and Keramane, A. (1996). Limit cycles and their stability in a pas- sive bipedal gait. Proceedings of the 1996 IEEE International Conference on Robotics and Automation, pages 246–251. Goswami, A., Thuilot, B., and Espiau, B. (1998). A study of the passive gait of a compass- like biped robot: Symmetry and chaos. International Journal of Robotics Research, 17(12):1282–1301. Hobbelen, D. G. E. and Wisse, M. (2007). Limit cycle walking. In Hackel, M., editor, Humanoid Robots: Human-like Machines, pages 277–294. I-Tech Education and Publishing, Vi- enna, Austria. Ikemata, Y., Sano, A., and Fujimoto, H. (2006). A physical principle of gait generation and its stabilization derived from mechanism of fixed point. Proceedings of the 2006 Conference on International Robotics and Automation, pages 836–841. Kinugasa, T., Kotake, K., Haruki, T., Tanaka, H., and Yoshida, K. (2008). 3D passive dynamic walker with sprung ankle and flat foot -a design method by natural frequency in- dex without yaw and roll compensator Proceedings of the 2008 JSME Conference on Robotics and Mechatronics, pages 1P1–B12–. Kuo, A. D. (1999). Stabilization of lateral motion in passive dynamic walking. International Journal of Robotics Research, 18(9):917–930. McGeer, T. (1990). Passive dynamic walking. International Journal of Robotics Research, 9(2):62– 82. McGeer, T. (1993). Passive dynamic biped catalogue, 1991. In Chatila, R. and Hirzinger, G., editors, Experimental Robotics II: The 2nd International Symposium, Toulouse, France, June 25-27 1991, pages 465–490. Springer-Verlag. McMahon, T. A. (1984). Mechanics of locomotion. International Journal of Robotics Research, 3(2):4–28. Narukawa, T., Yokoyama, K., Takahashi, M., and Yoshida, K. (2008). A simple 3D straight- legged passive walker with flat feet and ankle springs. IEEE/RSJ International Confer- ence on Intelligent Robots and Systems, pages 2952–2957. Narukawa, T., Yokoyama, K., Takahashi, M., and Yoshida, K. (2009a). Design and construction of a simple 3D straight-legged passive walker with flat feet and ankle springs. JSME Journal of System Design and Dynamics, 3(1):1–12. Narukawa, T., Takahashi, M., and Yoshida, K. (2009b). Design and Stability Analysis of a 3D Rimless Wheel with Flat Feet and Ankle Springs. JSME Journal of System Design and Dynamics, 3(3):258-269. Schwab, A. L. and Wisse, M. (2001). Basin of attraction of the simplest walking model. Pro- ceedings of the ASME Design Engineering Technical Conference, pages DETC2001/VIB– 21363–. Tedrake, R., Zhang, T. W., Fong, M F., and Seung, H. S. (2004). Actuating a simple 3D pas- sive dynamic walker. Proceedings of the IEEE International Conference on Robotics and Automation, 2004(5):4656–4661. Tedrake, R. L. (2004). Applied Optimal Control for Dynamically Stable Legged Locomotion. PhD thesis, Massachusetts institute of technology. Wisse, M., Hobbelen, D. G. E., Rotteveel, R. J. J., Anderson, S. O., and Zeglin, G. J. (2006). Ankle springs instead of arc-shaped feet for passive dynamic walkers. Proceedings of the IEEE-RAS International Conference on Humanoid Robots, pages 110–116. Wisse, M. and Schwab, A. L. (2005). Skateboards, bicycles, and three-dimensional biped walk- ing machines: Velocity-dependent stability by means of lean-to-yaw coupling. Inter- national Journal of Robotics Research, 24(6):417–429. Wisse, M., Schwab, A. L., van der Linde, R. Q., and van der Helm, F. C. T. (2005). How to keep from falling forward: Elementary swing leg action for passive dynamic walkers. IEEE Transactions on Robotics, 21(3):393–401. Wisse, M., Schwab, A. L., and vander Linde, R. Q. (2001). A 3D passive dynamic biped with yaw and roll compensation. Robotica, 19:275–284. Wisse, M. and van Frankenhuyzen, J. (2003). Design and construction of mike; a 2D au- tonomous biped based on passive dynamic walking. Proceedings of the Second In- ternational Symposium on Adaptive Motion of Animals and Machines, pages 4–8. CuttingEdgeRobotics2010144 ActiveKnee-releaseMechanismforPassive-dynamicWalkingMachines 145 ActiveKnee-releaseMechanismforPassive-dynamicWalkingMachines KalinTrifonovandShujiHashimoto X Active Knee-release Mechanism for Passive-dynamic Walking Machines Kalin Trifonov and Shuji Hashimoto Waseda University Japan 1. Introduction In this chapter we will present the design and development of two knee mechanisms. One uses permanent magnets to lock the knee in its extended position and the other features an active mechanism for releasing the passively locked knee. We will also present a comparison between the experimental results achieved with each of the two knee mechanisms. One of the big, and still unsolved, problems in robotics is achieving efficient and stable bipedal walking. There are two main strategies used to control walking. First, the traditional approach is to control the joint-angle of every joint at all times. Crucial disadvantages of this approach are that it results in a non-efficient gait in terms of energy consumption (Collins et al., 2005), it requires complex controllers and programming, and this strategy often results in gaits that are unnatural when compared to the human gait. Second is a somewhat new strategy called passive-dynamic walking, introduced by Tad McGeer (McGeer 1990) in the late 80’s, early 90’s. A walker based on the passive-dynamic walking principle uses its own mechanical dynamics properties to determine its movement. Such walkers can walk down slight inclines without any actuators, sensors or controllers. The energy that is necessary in order to sustain the walking motion is provided by gravity. The force of gravity is also enough to offset the losses due to the impact of the feet on the ground and friction. The advantages of passive-dynamic walking are high-energy efficiency, simple or no control, and a human-like gait. The main disadvantage is that because they are not actively powered, they can only walk on downhill slopes. This disadvantage can be eliminated by modifying walkers to include actuators that supply the necessary power instead of gravity (Collins and Ruina, 2005; Wisse and Frankenhuyzen, 2003; Wisse, 2004). This enables them to walk not only downhill, but on level and uphill surfaces as well. This possibility greatly increases the prospects for practical application. The knee mechanism is a major part in passive-dynamic walkers. There are several different designs that have been implemented in walkers up to now. The original walker built by McGeer uses a mechanism with suction cups that keeps the knee extended as shown in Figure 1. The drawback of the suction cups design is that it is difficult to set up and not very efficient. 9 CuttingEdgeRobotics2010146 Fig. 1. Knee design with suction cups knee-locking. Another popular design is used in the University of Delft’s Mike (Wisse and Frankenhuyzen, 2003) and subsequent walkers Max and Denise (Wisse, 2004). The locking of the knee is achieved actively by McKibben muscles, which are counteracted by weak springs as shown in Fig.2. As a drawback we can mention that the McKibben muscles are not linear, and require controller that takes this feature into account. They also require a source of air. Fig. 2. Knee design with McKibben muscles knee-locking. A third popular knee design is implemented in the Cornell powered biped (3). It features an electromagnetic release system shown in Fig.3, where (A), (B), (C), (D), (E), (F), (G) and (H) are a latch arm, a roller, a shank, a hinge, a shaft, a latch surface, a thigh and a solenoid, respectively. This design is robust and easy to control, but it is comprised of many parts, which makes it quite complicated. A similar design, where an electromagnetic clutch is used to engage or disengage a knee motor is developed by Baines (Baines, 2005). Fig. 3. Knee design with electromagnetic knee-release. We developed our two knee locking mechanism with simplicity in mind. We wanted to understand if it was possible to develop a passive walker and a knee mechanism specifically based only on observation and experimentation without any modeling and simulations. A detailed model describing the mechanisms of generation and stabilization of a fixed point of passive walking, as well as leg-swing motion analysis of a passive-dynamic walker can be found in the research done by Prof. Sano’s team at the Nagoya Institute of Technology (Ikemata et al., 2007), (Ikemata et al., 2008). Our aim was to build a mechanism that is simple, robust, and easy to use and set up. The purposes of this chapter are to present the mechanical design of the two knee mechanisms, to introduce the achieved experimental results, to make a comparison between them, and to discuss their effectiveness. 2. Knee Mechanism with Permanent Magnets The dynamics of passive-dynamic walkers cause the swinging leg to bend and extend on its own. However, in order to achieve a stable gait, the knee must be able to swing with minimal friction, meaning minimal energy loss. Taking this into consideration, the most logical choice for the knee joint is a ball bearing. Additionally, the knee must be equipped with a knee-locking mechanism that supports the knee during its extended phase and prevents it from bending while bearing the weight of the walker. For our walker, the knee is cut from an aluminum block and is comprised of only an upper knee, to which the aluminum lower leg is attached directly through a shaft and a pair of ball ActiveKnee-releaseMechanismforPassive-dynamicWalkingMachines 147 Fig. 1. Knee design with suction cups knee-locking. Another popular design is used in the University of Delft’s Mike (Wisse and Frankenhuyzen, 2003) and subsequent walkers Max and Denise (Wisse, 2004). The locking of the knee is achieved actively by McKibben muscles, which are counteracted by weak springs as shown in Fig.2. As a drawback we can mention that the McKibben muscles are not linear, and require controller that takes this feature into account. They also require a source of air. Fig. 2. Knee design with McKibben muscles knee-locking. A third popular knee design is implemented in the Cornell powered biped (3). It features an electromagnetic release system shown in Fig.3, where (A), (B), (C), (D), (E), (F), (G) and (H) are a latch arm, a roller, a shank, a hinge, a shaft, a latch surface, a thigh and a solenoid, respectively. This design is robust and easy to control, but it is comprised of many parts, which makes it quite complicated. A similar design, where an electromagnetic clutch is used to engage or disengage a knee motor is developed by Baines (Baines, 2005). Fig. 3. Knee design with electromagnetic knee-release. We developed our two knee locking mechanism with simplicity in mind. We wanted to understand if it was possible to develop a passive walker and a knee mechanism specifically based only on observation and experimentation without any modeling and simulations. A detailed model describing the mechanisms of generation and stabilization of a fixed point of passive walking, as well as leg-swing motion analysis of a passive-dynamic walker can be found in the research done by Prof. Sano’s team at the Nagoya Institute of Technology (Ikemata et al., 2007), (Ikemata et al., 2008). Our aim was to build a mechanism that is simple, robust, and easy to use and set up. The purposes of this chapter are to present the mechanical design of the two knee mechanisms, to introduce the achieved experimental results, to make a comparison between them, and to discuss their effectiveness. 2. Knee Mechanism with Permanent Magnets The dynamics of passive-dynamic walkers cause the swinging leg to bend and extend on its own. However, in order to achieve a stable gait, the knee must be able to swing with minimal friction, meaning minimal energy loss. Taking this into consideration, the most logical choice for the knee joint is a ball bearing. Additionally, the knee must be equipped with a knee-locking mechanism that supports the knee during its extended phase and prevents it from bending while bearing the weight of the walker. For our walker, the knee is cut from an aluminum block and is comprised of only an upper knee, to which the aluminum lower leg is attached directly through a shaft and a pair of ball CuttingEdgeRobotics2010148 bearings (Trifonov and Hashimoto, 2006). For the locking mechanism, we are using a knee plate spacer and a knee plate, cut from acrylic, as with the original McGeer design, but we decided to try a new approach by using magnets instead of a suction cup. We adjust the locking magnetic force by changing the distance between the magnet(s) and the steel plate. This can be achieved either by using magnets with different sizes or by using a different number of magnets. The smaller the distance is, the stronger the force. Another advantage of the magnetic lock is that it does not require physical contact between the locking parts (magnet and steel plate). In this way the material wear is reduced and the lock can be used longer without having to worry about replacing some of its parts. 3D renderings are shown in Fig. 4, where (A) is knee, (B) is knee plate, (C) is magnet(s), and (D) is a steel plate. A drawing of the knee mechanism with some main dimensions is shown in Fig. 5. Fig. 4. 3D renderings of the knee mechanism with permanent magnets. Fig. 5. Drawing of the knee mechanism with permanent magnets. 3. Knee Mechanism with an Active Release System We designed a second newer, simpler, and lower in weight knee-locking mechanism (Trifonov and Hashimoto, 2007). The locking mechanism is constructed of acrylic, ABS, steel, and aluminum. The knee-locking mechanism consists of a knee (A), knee plate (B), locking axle (C), locking hook (D), base plate (E), and a DC motor (F) as shown in Fig. 6 and Fig. 7. Additionally, there is a switch attached to each foot of the walker, which is used to control the DC motor, but is not shown in the figure. The entire knee mechanism was designed in 3D modeling software and cut on a CAM machine. The knee is cut from aluminum, the knee plate from acrylic, the locking axle from steel, and the locking hook and the base plate are cut from ABS. An active release system has been implemented before on a passive-dynamic walker. The Cornell powered biped (Collins and Ruina, 2005) uses an electromagnetic solenoid for the release of the passively locked knee mechanism. The advantages of our system are the much simpler design and the absence of a controller. The locking action is done passively. As the swing leg extends before hitting the ground, the locking axle hits the front edge of the locking hook, lifting it. After the locking axle passes under the hook, it comes back down to lock the axle, effectively locking the knee itself. The locking hook is balanced by a counter weight in such a way that it comes back down to its initial position after the locking axle has lifted it. Just before the stance leg lifts from the ground and starts to swing, the foot switch comes into contact with the ground and switches to the ON position, thus turning on the power for the DC motor. This causes the motor to lift the locking hook and release the knee. Immediately after the leg lifts off the ground and starts swinging, the foot switch returns to the OFF position, cutting the power, and the locking hook returns to its initial position. The foot switch is mounted to the side of the foot plate, such that it does not influence the walking of the machine. Fig. 6. 3D renderings of the knee mechanism with an active release. ActiveKnee-releaseMechanismforPassive-dynamicWalkingMachines 149 bearings (Trifonov and Hashimoto, 2006). For the locking mechanism, we are using a knee plate spacer and a knee plate, cut from acrylic, as with the original McGeer design, but we decided to try a new approach by using magnets instead of a suction cup. We adjust the locking magnetic force by changing the distance between the magnet(s) and the steel plate. This can be achieved either by using magnets with different sizes or by using a different number of magnets. The smaller the distance is, the stronger the force. Another advantage of the magnetic lock is that it does not require physical contact between the locking parts (magnet and steel plate). In this way the material wear is reduced and the lock can be used longer without having to worry about replacing some of its parts. 3D renderings are shown in Fig. 4, where (A) is knee, (B) is knee plate, (C) is magnet(s), and (D) is a steel plate. A drawing of the knee mechanism with some main dimensions is shown in Fig. 5. Fig. 4. 3D renderings of the knee mechanism with permanent magnets. Fig. 5. Drawing of the knee mechanism with permanent magnets. 3. Knee Mechanism with an Active Release System We designed a second newer, simpler, and lower in weight knee-locking mechanism (Trifonov and Hashimoto, 2007). The locking mechanism is constructed of acrylic, ABS, steel, and aluminum. The knee-locking mechanism consists of a knee (A), knee plate (B), locking axle (C), locking hook (D), base plate (E), and a DC motor (F) as shown in Fig. 6 and Fig. 7. Additionally, there is a switch attached to each foot of the walker, which is used to control the DC motor, but is not shown in the figure. The entire knee mechanism was designed in 3D modeling software and cut on a CAM machine. The knee is cut from aluminum, the knee plate from acrylic, the locking axle from steel, and the locking hook and the base plate are cut from ABS. An active release system has been implemented before on a passive-dynamic walker. The Cornell powered biped (Collins and Ruina, 2005) uses an electromagnetic solenoid for the release of the passively locked knee mechanism. The advantages of our system are the much simpler design and the absence of a controller. The locking action is done passively. As the swing leg extends before hitting the ground, the locking axle hits the front edge of the locking hook, lifting it. After the locking axle passes under the hook, it comes back down to lock the axle, effectively locking the knee itself. The locking hook is balanced by a counter weight in such a way that it comes back down to its initial position after the locking axle has lifted it. Just before the stance leg lifts from the ground and starts to swing, the foot switch comes into contact with the ground and switches to the ON position, thus turning on the power for the DC motor. This causes the motor to lift the locking hook and release the knee. Immediately after the leg lifts off the ground and starts swinging, the foot switch returns to the OFF position, cutting the power, and the locking hook returns to its initial position. The foot switch is mounted to the side of the foot plate, such that it does not influence the walking of the machine. Fig. 6. 3D renderings of the knee mechanism with an active release. CuttingEdgeRobotics2010150 Fig. 7. Drawing of the knee mechanism with an active release. 4. Experiments and Results To compare the two knee mechanisms, experiments were conducted with the same walker shown in Fig. 8, built from square aluminum tubes for the legs and 2mm thick steel plate for the feet (Trifonov and Hashimoto, 2006). For the thighs and lower legs, we used 2.5 by 2.5cm square aluminum tubes with lengths of 34 and 43.5cm respectively. The total height of the walker is 89cm and the radius of the feet is 12.3cm. The total weight is 4.5kg. The knees were outfitted first with the magnetic system and then with the active release one. The walker was set on a ramp, which measures 3m in length, 90cm in width, and has a 3 grade relative to the ground. The ramp is covered with a rubber mat to reduce the chance of foot slippage. We performed several sets of a hundred trials (walks) down the ramp for both knee mechanisms and counted the steps that the walker completed each time. We denote a trial as successful if the walker manages to make five to seven steps before it exits the ramp. While five to seven steps may seem short, we postulate that after five steps, the walker has achieved a steady gait, and would ideally continue assuming a longer ramp existed. However, the impracticality of a longer ramp led us to set this number of steps as the criteria for deciding walk success. Fig. 9 shows a comparison between the two knee mechanism designs in terms of average number of steps made in each of the hundred trials. As the results show, using the knee mechanism with active release, we can achieve a reasonable amount of successful trials. Out of a hundred trials, the walker achieved an average of forty-four successful walks with the active release system, while the magnetic approach resulted in only seven. In addition, using the active mechanism produces fewer failures than the magnetic one. Fig. 8. Walker on the ramp, outfitted with knees with active release system. Fig. 9. Comparison between the experimental results achieved with the two mechanisms. [...]... (-s5c6s7 + c5c7 )s8 )s9 + (-(s5c6c7 + c5s7 )s8 + (-s5c6s7 + c5c7 )c8 )c9 (14) s 2 z -(-s 6 c 7 c 8 + s 6 s 7 s 8 )s 9 + (s 6 c 7 s 8 + s 6 s 7 c 8 )c 9 (15) 160 Cutting Edge Robotics 2010 a 2 x c 5s 6 ( 16) a 2 y s 5s 6 (17) a2z c6 (18) a2z c6 (19) P2 x ((c5c6c7 - s5s7 )c8 + (-c5c6s 7 - s 5c7 )s8 )c9 L7 + (-(c5c6c7 - s5s 7 )s8 + (-c5c6s 7 - s5c7 )c8 )s9 L7 (20) + (c5c6c7 - s 5s 7 )c8 L6 +... (c5c6c7 - s 5s 7 )c8 L6 + (-c5c6s7 - s5c7 )s8L6 + (c5c6c7 - s5s 7 )L5 + c5L 4 P2 y ((s5c6c7 + c5s7 )c8 + (-s5c6s7 + c5c7 )s8 )c9L7 + (-(s5c6c7 + c5s7 )s8 + (-s5c6s7 + c5c7 )c8 )s9L7 + (s5c6c7 + c5s7 )c8L6 + (-s5c6s7 + c5c7 )s8L6 + (s5c6c7 + c5s7 )L5 + s5L4 P2 z (-s 6 c 7 c 8 s 6 s 7 s 8 )c 9 L 7 (s 6 c 7 s 8 s 6 s 7 c 8 )s 9 L 7 - (s 6 c 7 c 8 s 6 s 7 s 8 )L 6 - s 6 c 7 L 5 (21) (22) 2.1.3 Forward... + (-c 5 c 6 s 7 - s 5 c 7 )c 8 )s 9 (10) n 2 y ((s 5 c 6 c 7 + c 5s 7 )c 8 + (-s 5 c 6 s 7 + c 5 c 7 )s 8 )c 9 (-(s 5 c 6 c 7 + c 5s 7 )s 8 + (-s 5 c 6 s 7 + c 5 c 7 )c 8 )s 9 (11) n 2 z (-s 6 c 7 c 8 + s 6 s 7 s 8 )c 9 + (s 6 c 7 s 8 + s 6 s 7 c 8 )s 9 (12) s2 x -((c5c6c7 - s5s 7 )c8 + (-c5c6s 7 - s5c7 )s8 )s9 + (-(c5c6c7 - s5s 7 )s8 + (-c5c6s 7 - s5c7 )c8 )c9 (13) s2 y -((s5c6c7 + c5s7... shown in Figure 4 u J2 H ( 46) r2 norm ( J 2 ) (47) r3 norm (u ) (48) With vectors u, r2, r3 and lengths L4, L5, L6 and using the law of cosines the angles β4, β5 and 6 are obtained L2 r32 r22 4 2 L4 r3 (49) L2 r32 L2 5 6 2 L5r3 (50) L2 L2 r32 6 5 2 L6 L5 (51) 4 a cos 5 a cos 6 a cos 166 Cutting Edge Robotics 2010 MCP flexion/extension... described in equation (65 ) R , MCP - _ fe , MCP _ fe _ fe R , MCP 7 L , MCP 12 _ fe (66 ) _ fe _ fe 60 º (67 ) L , MCP _ fe 50 º (68 ) M , MCP Ring and little abduction/adductions are similar In most cases is very difficult to change the abduction of a finger without changes in the other Equation (69 ) represents this constraint R , MCP _ aa L , MCP _ aa - (65 ) Flexion of middle and... MCP a tan 2[ x 2 , x1 ] (60 ) Auxiliary variables are calculated such as: x3 ( L1 ( L2 x1 )) x4 L2 x2 J xy J12x J12y x5 x6 J 1z x4 J xy x3 (61 ) 2 2 x3 x4 J 1 z x3 J xy x4 2 2 x3 x4 TMC_fe is obtained as: T , TMC a tan 2[ x 6 , x 5 ] (62 ) a tan 2[ n1 z , n12x n12y ] (63 ) _ fe Finally, IP is obtained as: T , IP T , TMC _ fe T , MCP (64 ) 3 Main Constraints of... 0 1 cT ,TMC _ fe 0 1 0 1 T0 (uThumb ) 0 0 0 0 0 1 LT , D c T , IP LT , D s T , IP 0 1 0 uT , x 0 uT , y 1 uT , z 0 1 (25) ( 26) (27) (28) 162 Cutting Edge Robotics 2010 n1 x n 0 T4 1 y n1 z 0 s1 x a1 x s1 y s1 z a1 y a1 z 0 0 P1 x P1 y P1 z 1 (29) Where: n1 x (c 1 c 2 c 3 - c 1s 2 s 3 )c 4 + (-c 1 c 2 s 3 - c 1s... walking International Journal of Robotics Research, 9(2) pp .62 82, (1990) K Trifonov, S Hashimoto, Design Improvements in Passive-Dynamic Walkers, In Proc International Conference "Automatics and Informatics ' 06" , Sofia, Bulgaria, pp 3538, (20 06) K Trifonov, S Hashimoto, Active knee-lock release for passive-dynamic walking machines, In Proc IEEE Robio 2007, pp 958- 963 , Sanya, China, (2007) M Wisse,... similar and proportional to the ring flexion as described in equation (67 ) and (68 ) R , MCP - 2 M 3 Flexion in the ring finger θR,MCP_fe is produced when there is flexion solely in the little finger It is described in equation (66 ) A small flexion is also produced in the middle when flexion in the little is high R , MCP - _ fe (69 ) Finally, another involuntary movement appears in the middle flexion... 1 0 0 1 (6) 0 Li , D c i , DIP 0 Li , D s i , DIP 1 0 0 1 (7) 0 0 ui,x 1 0 ui, y 0 1 ui, z 0 0 1 1 0 1 T0 (u i ) 0 0 n2 x n 0 T5 2 y n2 z 0 0 Li , p c i , MCP _ fe 0 Li , p s i , MCP _ fe 1 0 0 1 (8) P2 x P2 y P2 z 1 (9) Where: n2 x ( (c 5 c 6 c 7 - s 5s 7 )c 8 + (-c 5 c 6s 7 - s 5 c 7 )s 8 )c 9 (-(c 5 c 6 c 7 - s 5s . 98 768 769 8 768 762 )ccss+sc(s+)ssss+cc-(-s z s (15) Cutting Edge Robotics 2010 160 65 2 sc x a ( 16) 65 2 ss y a (17) 62 c z a (18) 62 c z a (19) 45575 765 6875 765 6875 765 79875 765 875 765 79875 765 875 765 2 Lc+)Lss-cc(c+L)scs-sc(-c+L)css-cc(c+ L)s)ccs-sc(-c+)sss-cc(-(c+L)c)scs-sc(-c+)css-cc((c x P . 45575 765 6875 765 6875 765 79875 765 875 765 79875 765 875 765 2 Lc+)Lss-cc(c+L)scs-sc(-c+L)css-cc(c+ L)s)ccs-sc(-c+)sss-cc(-(c+L)c)scs-sc(-c+)css-cc((c x P (20) 45575 765 6875 765 6875 765 79875 765 875 765 79875 765 875 765 2 Ls+)Lsc+cc(s+L)scc+sc(-s+L)csc+cc(s+ L)s)ccc+sc(-s+)ssc+cc(-(s+L)c)scc+sc(-s+)csc+cc((s y P . 9875 765 875 765 9875 765 875 765 2 )s)ccs-sc(-c+)sss-cc(-(c)c)scs-sc(-c+)css-cc(c( x n (10) 9875 765 875 765 9875 765 875 765 2 )s)ccc+sc(-s+)ssc+cc(-(s)c)scc+sc(-s+)csc+cc((s y n (11) 98 768 769 8 768 762 )scss+sc(s+)csss+cc(-s z n
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