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Proceedings of the 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics Monterey, California, USA, 24-28 July, 2005

A Robot with Cockroach Inspired Actuation and Control Jong-ung Choi, Brandon L. Rutter, Daniel A. Kingsley, Roy E. Ritzmann, Roger D. Quinn

Abstract— Robot V has been constructed with inspiration from the death head cockroach, Blaberus discoidalis. Its relative leg segment lengths, joint degrees of freedom, exoskeleton structure, relatively light legs, and location of its center of mass are all similar to those of the cockroach. In an attempt to take further advantage of the neuromechanics of the animal, actuators with muscle-like properties have been employed. The robot’s controller includes biologically inspired gait generation and inverse kinematics components. An actuator tensioning reflex which approximates the function of muscle tone is introduced, and resulting improvements to system response are shown.

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I. INTRODUCTION

OR many years roboticists have been attempting to develop a fully autonomous robot with animal-like agility provided by multi-segmented legs. Such a machine could be very useful, but development has been hampered by actuators, power technology, and control schemes that are inadequate compared with those of animals. Biology provides a wealth of inspiration for legged robot design. There are millions of species of animals that have evolved effective solutions for legged locomotion. Insects in particular are exemplary not only for their speed and agility but also for their ability to traverse some of the most difficult terrains imaginable. There are mechanisms other than legs capable of producing land-based locomotion, most notably wheels and treads. While these devices are easier to design and implement than legs, they have disadvantages that limit their mobility and survival capability in the real world. Neither wheels nor treads are capable of traversing terrain as complex as that readily negotiated by legged animals [1]. Even wheeled and tracked vehicles designed specifically for harsh terrains cannot maneuver over an obstacle significantly taller than themselves. A legged vehicle, however, might be expected to climb an obstacle up to twice its own height, much like a cockroach can [2]. In any environment without fairly flat, continuous terrain, a walking vehicle is preferable to a wheeled or tracked one. For example, legged vehicles are far more capable of

Manuscript received March 8, 2005. This research is funded by the Ohio Space Grant Consortium, NSF IGERT DGE-9972747 and Eglin AFB grant F08630-03-1-0003. J. Choi and B. L. Rutter are with the Biorobotics Laboratory at Case Western Reserve University, Cleveland, OH 44106 USA (phone: 216-368-5216; e-mail: jxc91@case.edu and blrutter@case.edu). Dan Kingsley completed his Ph.D. research at the Case Biorobotics Lab in 2004 and is working for Sarcos Research Ltd (d.kingsley@sarcos.com). R. E. Ritzmann is Professor of Biology at Case (rer3@case.edu). R. D. Quinn is Professor of Mechanical and Aerospace Engineering at Case (rdq@case.edu) and the Director of the Biorobotics Lab.

0-7803-9046-6/05/$20.00 ©2005 IEEE.

navigating an intermittent substrate—such as a slatted surface—than wheeled vehicles [3]. Though legs have many advantages, they have the disadvantage of requiring more complex control. Biologically inspired mechanics and control can benefit each other, and using both can help address the legged locomotion control problem. In animals, the mechanical and control systems are functionally combined into neuromechanical systems. Their mechanical systems can reject disturbances that are too rapid for their control systems to process [4], while their neural control systems tune their mechanics for efficient locomotion [5]. A well-designed legged robot should share these features. II. MECHANICS A. Actuator Selection The selection of actuators plays a pivotal role in any mobile robot design, as the shape, size, weight, and strength of an actuator must all be taken into account. The power source for the actuators often provides the greatest constraint on a robot’s potential abilities. Biological organisms have a great advantage over mechanical systems in that muscle has a favorable force-to-weight ratio and effective energy storage mechanisms. Muscle’s tunable passive compliance is also well-suited for energy-efficient legged locomotion. The most frequently used actuators—electric motors and pneumatic/hydraulic cylinders—are far from equivalent to their biological counterparts. While electric motors are readily available in a wide range of sizes and are generally easy to control, these devices have several disadvantages. Most importantly, their force-to-weight ratio is far lower than that of other common actuation devices, which makes them unsuitable for many applications. Typically, electric systems have a power-to-weight ratio of 50-100 W/kg

Fig. 1. Robot V can stand and walk with no feedback control.

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(including only motor and gear reducer), whereas pneumatic and hydraulic systems produce 100-200 W/kg (including actuator and valve mass) [6] and biological muscle produces 40-250 W/kg [7]. Braided pneumatic actuators (BPAs) have a favorable power-to-weight ratio (1-1.5 kW/kg) [8], provide a number of advantages over conventional actuation devices, and share some important characteristics with biological muscle. These devices consist of an expandable fiber mesh wrapped around an inflatable bladder. The resulting actuator is significantly lighter than a standard air cylinder, but is capable of producing greater forces than its heavier counterpart. When the bladder is filled with pressurized air, its volume tends to increase. Because of the constant length of the mesh fibers, the actuator expands radially while simultaneously contracting along its axis (or contracts radially while extending, depending on the fiber mesh angle). The result is a contraction that produces a force-length curve akin to the rising phase of muscle [9]. An important property of contractile BPAs is that they produce zero force at maximum contraction and produce maximum force when fully extended. Therefore, similar to muscle, the force output of these actuators is self-limiting by nature. A braided pneumatic actuator driven by an unstable controller is less likely to damage itself or the surrounding structure. Because of this property, BPAs are well-suited for implementation of positive load feedback, which is used by animals such as cockroaches, cats, and humans [10]. Like biological muscle, BPAs only exert force in one direction and must be used in opposition to an antagonist, which is usually another actuator. This property is significant for useful application: although it requires the use of two actuators or sets of actuators at each joint, this animal-like configuration allows co-contraction, which can be used for stiffness control. Robot V uses a type of BPA made by Festo Inc. in which the fiber mesh is molded into the rubber tube [11]. This BPA has a longer fatigue life, but has the disadvantages of being very stiff in extension past the resting length and requiring a higher inflation pressure. B. Mechanical Design The design of Robot V is inspired by the death head cockroach Blaberus discoidalis. It is not yet feasible to capture the full range of motion exhibited by the insect, up to seven degrees of freedom (DOF) per leg. Analysis of leg motion during locomotion suggests that this is unnecessary, however, because in many cases a few DOF produce the majority of a leg’s movement. Three DOF in the rear legs, four in the middle legs, and five in the front legs are sufficient to produce cockroach-like climbing and walking movements [12],[13]. The different number of DOF used in each set of limbs represents the task-oriented nature of each pair of legs that is seen in most legged animals [14],[15]. On the insect, the front legs are relatively small and weak, but highly dexterous. This dexterity is attained in the robot through three DOF between the body and coxa (Figure 2). These DOF are referred to (from most proximal to most distal) as

, with an axis parallel to the intersection of the median and coronal planes (in the z direction); , with an axis parallel to the median and transverse planes (in the y direction); and , with an axis parallel to the coronal and transverse planes (in the x direction). The middle legs on the insect play an important role in turning and climbing (rearing) [2], but they sacrifice some dexterity for power. On Robot V, the middle legs have only two degrees of freedom ( and ) between the body and coxa. Finally, the cockroach uses its rear legs primarily to push itself forward, These legs are larger, much more powerful and make only limited use of the body-coxa joint [13]; likewise, the body-coxa joints of the robot’s rear legs have only the  DOF.

Fig. 2. Schematic of the front right leg of Robot V, illustrating its five degrees of freedom.

Fully assembled, Robot V weighs 15 kilograms and is one meter long by half a meter wide: twenty times the size of the animal. Despite the large difference in scale, the robot’s design is similar to the animal’s in many ways beyond its joint DOF and muscle-like actuators. The relative length of its leg segments are proportioned similar to those of the animal, and its legs are relatively light compared to its body because of their exostructure design. The actuator control valves are placed on the body such that the center of mass is at the body-coxa joint of the rear legs, as in the animal. The passive ranges of motion of the joints were measured by moving each joint by hand and were found to be equal to or greater than those that the animal uses in walking and climbing [2],[5],[12]. III. CONTROL A RCHITECTURE Though progress has been made on various fronts [16], [17], the best method of controlling a legged robot with multiple kinematically redundant limbs remains an open question. Additionally, it is interesting and often informative to try implementing biological control schemes, but the data available are usually limited to only a subset of those necessary to control an entire robot [18]. For these reasons, a flexible and extensible control architecture has been

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implemented in the RTLinux-gpl [19],[20] real-time operating system. The RTLinux system architecture imposes a separation of control into three regimes. At the lowest level is the robot hardware itself: passive properties of the BPAs allow spring-like rejection of perturbation and mechanical filtering of forces resulting from non-smooth inlet pressures. These processes take place at the speed of physics—similar to behavior apparent in cockroaches and other animals [4],[15]. Different leg and body postures and differing levels of co-contraction in the actuators should be capable of changing the dynamics of this physical level of control to suit varying demands in different situations. At the middle level are the real-time (RT) control threads. These include direct control of the valves and joint-level control, and will be extended to include leg coordination and possibly higher-level posture control and immediate path planning. Any control task which must be executed with a certain priority or frequency resides in this level. At the highest level, in the standard Linux user space, are the user interface, communication, data logging, and configuration. It is also possible to carry out nontime-critical planning tasks or learning at this level. The separation of various control tasks in the RT (middle) level is the most interesting part here: the control data and algorithmic implementation are set up such that each control component (joint, actuator, or valve) can have a unique control strategy implemented by a unique RT thread. For example, it is possible to easily test a new control strategy on one joint while the rest of the robot, including components associated with that joint, is being handled by an older, perhaps better characterized, control process. This method was used while developing the reflex described in Section IV below. A valve output module that drives valves with 50 Hz pulse width modulation and a joint position controller have been implemented in the RT domain. The leg control and coordination modules are not currently implemented in the RT process domain, but the joint control module is capable of following a “puppet file” containing either desired angles or direct, non-feedback valve commands. Leg and coordination control are currently implemented on a dynamic model of the robot, and the desired angles generated are fed to the robot in an angle puppet file. This is the method used for generating the desired angles and foot paths shown in Sections IV–VI. Direct valve command mode has been used to demonstrate the desirable passive properties of BPAs. The robot is able to reject perturbations using only the spring-like properties of the actuators without any form of active posture control. The robot can also walk forward slowly while following an open-loop valve puppet file. Although this is by no means the robust, agile walking that is the ultimate goal of this project, it is a clear demonstration of the robot’s capabilities and the advantages offered by BPAs. The ability to move using only an open loop controller

demonstrates the usefulness and effectiveness of the mechanical domain of control when BPAs and kinematically redundant limbs are used. IV. JOINT CONTROL A. Position Control The joint position control is based on the method of Colbrunn [21] for antagonistic pairs of braided pneumatic actuators. In this method the exhaust of an actuator is given the same command as the inlet of its antagonist, based on position feedback. The inlets and exhausts are then biased against each other based on stiffness feedback. On Robot V, however, the stiffness control component of this controller is not implemented, since our current stiffness model for the Festo actuators is not sufficient for this purpose. Even such a simple control algorithm produces fairly good results for position control under load of constant direction, probably due in part to the effective integral gain inherent in the trapped-air pneumatic actuators.

Fig. 3. Examples of improvements in joint control resulting from a tensioning reflex. The effects are most easily noticeable under the most variable loading conditions. For the femur-tibia joint these are walking movements with light ground contact, because of the ground discontinuities. For the  DOF it is air-walking, because it has to deal with the inertia of the entire leg itself, rather than relying on the ground for braking movement.

The minimally extensible Festo actuators require a resting length equal to their maximum length in order to get the desired range of motion from a joint. The problem of actuator slack is encountered if the position controller described above is used under varying loads. When an actuator under load is inflated, its antagonist is deflated. If the antagonist is deflated to the point of becoming slack and the load is then released, the pressurized muscle will pull the joint toward itself with very little resistance, and will drastically overshoot the desired angle before the previously loaded actuator can deflate, or before its antagonist can inflate to provide sufficient resistance. This behavior is shown in the “Without Reflex” data of Fig. 3. This problem was not observed in the more classic BPA design of the actuators from the Shadow Robot Company [22] used by

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Colbrunn, since those actuators were extensible and were pre-tensioned [23] so they would go slack less often and more gracefully. B. Tensioning Reflex The problem with actuator slack and the delay and deadband it causes has led us to develop a reflex to maintain tension in the Festo actuators. The current implementation uses data gathered from the robot offline to determine a relationship between joint angle and the minimum no-slack pressure for each actuator. A modified pressure controller running in coordination with the position controller then ensures that the pressure in an actuator is never below this minimum pressure. This reflex thus serves a purpose similar to the maintenance of muscle tone in animals, albeit with a much simpler method of implementation [24],[25]. Even without systematic tuning of the reflex parameters, angle tracking and dynamic performance under varying load, including the rejection of large disturbances by the physical level of control, are improved over the case with no reflex, as shown in Fig. 3. The repeatability of the controller during air walking is also improved, as shown in Fig. 4 and Fig. 5.

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amphibians [26]. Solving the inverse kinematics problem using the MMP method is a computationally intensive task. Also, contrary to most industrial robot applications, locomotion with compliant legs does not need highly accurate joint position. A neural network can therefore be trained to solve the inverse kinematics problem in a computationally efficient manner during walking. To provide a training set for the neural network, a cubic grid is defined and the points that the foot can reach without singularity are used to solve the inverse kinematics problem using the MMP method. We used fully connected feed-forward neural nets with a single hidden layer having a sigmoidal threshold function. The weights and biases were found using the MATLAB neural net toolbox to create a feed-forward network and train it using back propagation. The trained networks were then implemented in the controller for the dynamic model robot. Fig. 6 shows the structure of the neural networks used.

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Fig. 4. Desired (red) and actual (blue) foot paths of right front leg without (A) and with (B) tension reflex. Fig. 6. Topologies of inverse kinematics neural networks. Each neural network is trained on a set of inverse kinematics solutions spanning the leg workspace. Inputs are the foot positions and outputs are the joint angles.

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Fig. 5. Mean and variance of position errors of right front foot without (blue) and with (red) tensioning reflex.

V. LEG CONTROL A. Inverse Kinematics Solution The front and middle legs of Robot V are kinematically redundant in three dimensions (3D). Their redundancy imparts maneuverability and flexibility, but it is difficult to find inverse kinematics solutions because there may be infinitely many solutions corresponding to a desired 3D foot position. To find inverse kinematics solutions, the Modified Moore-Penrose (MMP) method is used. Mussa-Ivaldi and Hogan hypothesized that this is the solution used by

Fig. 7. (A) Workspaces of rear (left) middle (center) and front (right) legs, found using forward kinematics and active joint ranges of motion. (B) Desired foot paths (blue) vs. paths generated using joint angles generated by neural network (green). Rear, middle and front paths are shown with respect to the body-coxa joint.

B. Path planning and Implementation of NN Path planning is done inside the workspace. Even though the workspace is in 3D, the desired path is currently designed in 2D because the design of a 2D swing path is more convenient. Vertical planes parallel to the y-axis are defined at one-inch intervals through the workspace, and the plane with maximum possible trajectory length is chosen as

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the trajectory plane. The desired paths are plugged into the neural network, which provides the required joint angles. By using forward kinematics, we have verified that the joint angles produced by the neural network do indeed produce the desired path (Fig. 7B). Fig. 8 shows desired vs. actual joint angles for the left front leg, generated by the trained neural net and used to drive the proportional joint controllers during air walking. The system exhibits a time delay associated with the air transport necessary to pressurize and exhaust the actuators Gamma -40

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phase of neighboring legs receiving this signal. Mechanism 2 is a positive impulse that is sent when a leg completes its return stroke and enters stance. This motivates nearby legs to enter their swing by moving the PEP forward. In implementation, this mechanism was made to last for the first ten percent of the unmodified stance phase. Mechanism 5 is a positive ramp function that begins when a leg enters stance; thus, the further a leg is in stance, the more motivation there is for an adjacent leg to enter swing [27].

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Fig. 8. Desired (red) vs. actual (blue) joint angles of left front leg during air-walking.

VI. INTER-LEG COORDINATION One of the most robust biological models of insect leg coordination was proposed by Cruse and is based on the stick insect Carausius morosus [27]. This control model has been shown to be sufficiently robust to control previous robots built in the Case Biorobotics Lab. The gait control network consists of a node for each leg that communicates with all adjacent legs. Each leg carries out a basic step-and-swing motion between an anterior extreme position (AEP) and a posterior extreme position (PEP). The swing speeds of all the legs are the same under all circumstances. The stance speeds, however, while being the same between all legs, can be varied to produce different walking speeds. Cruse proposed six different types of signal mechanisms that could be relayed between legs, but previous work has shown that only three of these (labeled 1, 2 and 5) are needed to produce stable walking on irregular terrain [3]. During walking, each leg begins its stance phase at a preset and constant AEP and begins to move toward the preset PEP as stance continues. Signals received from adjacent legs act to adjust the PEP towards the anterior (positively) or posterior (negatively), and once a leg has reached its PEP, it will enter a swing phase and return to the AEP. Mechanism 1 outputs a constant negative value while a limb is in its return stroke. This has the effect of extending the stance

Although this algorithm is straightforward to implement, it is not guaranteed to produce a stable gait controller. It is necessary to determine scaling factors for the network mechanisms and to differentiate between ipsilateral (same side of body) and contralateral (opposite side of body) connections, since it has been found that for stable gait generation contralateral connections should be weaker than their ipsilateral counterparts [3]. Furthermore, different walking speeds will affect whether a given set of scaling factors will or will not produce stable walking. Through trial and error, the values in Table 1 were found, in simulation, to produce stable locomotion over a full range of walking speeds. TABLE 1: SCALING FACTORS FOR CRUSE CONTROLLER Ipsilateral Contralateral Mechanism 1 1.03 0.3 Mechanism 2 2.0 0.1 Mechanism 5 7.25 3.08 Cruse Gait

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Fig. 10. Cruse-generated tripod gait and foot paths for six legs. The numbers 1, 2,3,4,5 and 6 represent the left front, middle, rear and right front, middle, rear legs, respectively. All foot paths are represented with respect to the body-coxa joint.

Fig. 10 shows the tripod gait automatically generated by the Cruse controller. By simply changing the velocity, wave and tripod gaits can be generated. A foot path generator is also included in the robot’s gait controller. The swing path

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of middle leg is inspired by the corresponding cockroach foot path. This generates smoother walking than that observed with a parabolic foot path for the middle leg. Foot paths generated by the gait controller were used by the neural networks to produce joint angles, which were recorded and then followed using the proportional controller. Using this system Robot V is capable of coordinated walking movements while suspended in air, and walks forward when placed on the ground.

[9]

[10] [11] [12] [13]

VII. CONCLUSION Robot V has been constructed with leg designs similar to those of the cockroach Blaberus discoidalis. While standing, Robot V is able to resist perturbations and perform rudimentary walking with no sensory feedback because of the passive stiffness inherent in its artificial muscles. A Cruse-based gait controller and neural networks for inverse kinematics solutions produce joint angles necessary to move the feet as desired. Using the proportional controller to follow these joint positions, Robot V is capable of coordinated walking movements while suspended in air, and walks forward when placed on the ground. In air-walking, its joints follow their desired angles with a time shift that is caused by valve and air transport delays. The neural networks were trained offline to solve the inverse kinematics problems without intensive online calculations. The applied tensioning reflex prevents problems with actuator slack and reduces joint position errors. Future work will include force or stiffness control to improve walking.

[14] [15]

[16] [17]

[18] [19] [20] [21]

[22] [23]

ACKNOWLEDGMENT B. L. Rutter thanks Nick Barendt and Barry Drennan for their help taming RTLinux.

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M. H. Raibert and J. K. Hodgins, “Legged Robots,” in Biological Neural Networks in Invertebrate Neuroethology and Robotics, R. D. Beer, R. E. Ritzmann, and T. McKenna, 1993. J. T. Watson, R. E. Ritzmann, S. N. Zill, and A. J. Pollack, “Control of obstacle climbing in the cockroach, Blaberus discoidalis,” I. Kienmatics. J. Comp. Physiol. A, vol. 188, pp. 39—53, 2002. K. S. Espenschied, R. D. Quinn, H. J. Chiel, and R. D. Beer, “Biologically-Based Distributed Control and Local Reflexes Improve Rough Terrain Locomotion in a Hexapod Robot,” Robotics and Autonomous Systems, vol. 18, pp. 59—64, 1996. D. L. Jindrich and R. J. Full, “Dynamic stabilization of rapid hexapedal locomotion,” The Journal of Experimental Biology, vol. 205, no. 18, pp. 2803—2823, Sep. 2002. G. E. Loeb, I. E. Brown, and E. J. Cheng, “A hierarchical foundation for models of sensorimotor control,” Exp. Brain Res., vol. 126:, pp. 1—18, 1999. M. B. Binnard, “Design of a Small Pneumatic Walking Robot”, M.S. Thesis, M.I.T., Cambridge, MA, 1995. S. T. Davis and D. G. Caldwell, “The Bio-Mimetic Design of a Robot Primate Using Pneumatic Muscle Actuators,” in Proc. 4th International Conference on Climbing and Walking Robots (CLAWAR 2001), Karlsruhe, Germany, 24—26 Sept. 2001 D. G. Caldwell, G. A. Medrano-Cerda, and M. Goodwin, “Control of pneumatic muscle actuators,” IEEE Control Systems Magazine, vol. 12, no. 1, pp. 40—48, Feb. 1995.

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G. K. Klute, and B. Hannaford, “Modeling Pneumatic McKibben Artificial Muscle Actuators: Approaches and Experimental Results,” ASME Journal of Dynamic Systems, Measurements, and Control, submitted for publication Nov. 1998, revised Mar. 1999. A. Prochazka, D. Gillard, and D. J. Bennett, “Implications of Positive Feedback in the Control of Movement,” The Journal of Neurophysiology vol. 77, pp. 3237-3251, June 1997. Pneumatic Catalog 2004 – Pneumatic Muscle, Available: http://www.festo.com R. J. Bachmann, “A Cockroach-Like Hexapod Robot for Running and Climbing,” M.S. Thesis, Dept. Mech. & Aero. Eng., Case Western Reserve Univ., Cleveland, OH, 2000. J. T. Watson and R. E. Ritzmann, “Leg kinematics and muscle activity during treadmill running in the cockroach, Blaberus discoidalis: slow running,” J. Comp. Physiol. A, vol. 182. pp. 11—22, 1998. A. C. Powers, “Research in the Design and Construction of Biologically-Inspired Robots.” M.S. Thesis, Univ. of California, Berkeley, CA, 1996. R. E. Ritzmann, R.D. Quinn, and M.S. Fischer, “Convergent Evolution and Locomotion through Complex Terrain by Insects, Vertebrates and Robots,” Arth. Struct. Dev. vol. 33, no. 3, pp. 361-379, 2004. G. M. Nelson and R. D. Quinn, “Posture control of a cockroach-like robot,” IEEE Control Systems Magazine, vol. 19, no. 2, pp. 9—14, Apr. 1999. G. M. Nelson, “Learning About Control of Legged Locomotion Using A Hexapod With Compliant Pneumatic Actuators,” Ph.D dissertation, Dept. Mech. & Aero. Eng., Case Western Reserve Univ., Cleveland, OH, 2002. O. Ekeberg, M. Blumel, and A. Buschges, “Dynamic simulation of insect walking,” Arth. Struct. Dev., vol. 33, pp. 287—300, 2004. V. Yodiaken and M. Barabanov, “A Real-Time Linux,” in Proc. Linux Applications Development and Deployment Conf. (USELINUX), Anaheim, CA, 1997. RTLinux-gpl sources available: http://www.rtlinux-gpl.org/ R. W. Colbrunn, G. M. Nelson, and R. D. Quinn, "Design and Control of a Robotic Leg with Braided Pneumatic Actuators," in Proc. 2001 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS '01), Maui, HI 2001, vol. 2, pp. 992—998. Shadow Robot Company pages, Available: http://www.shadow.org.uk R. W. Colbrunn, “Design and control of a robotic leg with braided pneumatic actuators,” M.S. thesis, Dept. Mech. & Aero. Eng., Case Western Reserve Univ., Cleveland, OH, 2000. J. Sigwald, P. Raverdy, “Muscle tone.” In Handbook of Clinical Neurology, vol 1, P. J. Vinken, G. W. Bruyn, and R. Garcin, Eds. Amsterdam: North Holland, 1969, pp. 257—276. R. A. Davidoff, “Skeletal muscle tone and the misunderstood stretch reflex,” Neurology, vol. 42, no. 5, pp. 951—963, 1992. F. A. Mussa-Ivaldi and N. Hogan, “Integrable solutions of kinematic redundancy via impedance control,” Int. J. of Robotics Research, vol. 10, no. 5, pp. 481—491, Oct., 1991. H. Cruse, “What mechanisms coordinate leg movement in walking arthropods?” Trends in Neural Science, vol. 12, pp. 15—21, 1990.

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