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HEXAPOD - SIX-LEGGED WALKING ROBOT CONTROLLED WITH TODA-RAYLEIGH LATTICE

HEXAPOD - SIX-LEGGED WALKING ROBOT CONTROLLED WITH TODA-RAYLEIGH LATTICE Movement of six-legged robot is considered. The robot is built to conduct research on the control of different modes of walking with a particular focus on switching from one mode to another. The robot parameters are identified from real-time control experiments. A number of identical servo motors are responsible for the mobility of the hexapod. Therefore, in addition to the construction of hexapod and its control system and the ability to generate different gaits, it is important to create a simulation model of the hexapod including servo motors. The full state observer for the servo motor is an integral part of this model. Forward and backward kinematics of the robot legs is implemented. The six node Toda-Rayleigh lattice model is created and numerically simulated. The lattice is applied as the robot controller. Three different modes of robot gait are achieved. The algorithm used for the gait mode change is equipped with parameters that are experimentally chosen. KEYWORDS: walking robot, walking algorithm, modelling, simulation, identification 1. Introduction The hexapod walking robot equipped with a construction plate is considered. The robot moves through the movements of six identical legs. Every leg consists of three links driven by the Hitec HS-475HB servo motors. The leg construction is similar to the biological construction of the cockroach leg. Thanks to this fact the analysis of natural gaits of insects will be used during the control process design [1]. The robot is shown in Figure 1. The direct control layer with the preset controller in the feedback loop of the servo motor has been implemented by the servo manufacturer. Only the manufacturer is responsible for the tuning of servo controllers, no one else. The whole system includes of 18 servo motors. Figure 1. Photo of the hexapod, a six-legged walking robot. The simulation model of motion is built in Simulink. It is dedicated to the real-time control environment of the hexapod system [2]. The real environment consists of two layers: an FPGA circuit and a PC computer with the real-time operating system [3]. Thanks to six legs we can easily design walking algorithms such that the whole construction will be stable in every phase of walking (contrary to the four-legged and the two-legged robots). The walking algorithms have a very simple form. Usually they correspond to synchronously repetitive actions. In fact, these three algorithms correspond to three different gait modes. Figure 3 shows the way how the legs are moved in each gait mode. Figure 2. The robot leg. Figure 3. Scheme presenting three basic modes of the six-legged insect walk (L stands for left legs and R stands for right legs). In the case of the six-legged insects we consider three characteristic ways of walking: the insect moves one leg in a short period of time, the other five legs support the insect body the insect moves two legs in a short period of time, the other four legs support the insect body the insect moves three legs in a short period of time, the other three legs support the insect body In all simulations and experiments the plane walking surface without any imperfections is considered. The full cycle consists of the following modes: a) six stages, b) three stages and c) only two stages. The analysis of the insect motion helps to develop walking algorithms for the hexapod. One can conclude: at the same velocity of the single leg motion the robot can achieve three different velocities by changing the walk mode the robot can move forward or backward due to the applied sign of the control signal the higher number of legs raised, the faster the robot can move but the legs on which the robot is leaning are more burdened when a leg (or legs) are moving forward, the supporting legs have to move backward with the velocity given by the formula vb = v f n 6n (1) where v f is the velocity of the legs moving forward and n is the number of the legs moving forward. 2. Toda-Rayleigh Lattice (Ring) Cyclic behavior is well known in various scientific and engineering disciplines: in medicine, biology, biochemistry, robotics, etc. It is also important that individual elements of a system are identical and identically related. Therefore we can create spatial-temporal lattice (ring) models. For example, chemical compounds having atoms arrange in rings or closed-chain structures. Particularly, in the robotics discipline, for multi-leg walking robots different types of gait can be represented as the formation and propagation of waves modeled by an evolutionary ring, i.e. a lattice system. The TodaRayleigh lattice is used as a high level control algorithm. The dynamics of the system based on the lattice is described by equation (2). It consists of two main parts: the dissipative Toda lattice and Rayleigh-like energy pumping. If we imagine a chain of point masses coupled by Toda identical springs with dissipative effects then it becomes obvious that decaying nonlinear oscillations have to be stabilized by Rayleigh-type providing or taking away energy. It is interesting that the lattice-ring composed of six non-harmonically interacting devices permits to mimic standard hexapod gaits. The solutions of equation (2) are n - 1 limited cycles, so-called oscillatory modes and two non-oscillatory, rotary modes. The current stable state depends on external parameter pext 2 n = 0 e y yn1 yn yn yn+1 + f ( y 2 pext ) R yn f(x) = ( x 2 ) x (2) Equation (2) describes one node of the lattice (ring). Six identical nodes are connected as it is shown in Figure 4. It describes how the energy is passed over the ring. The parameter 0 is responsible for the frequency of the soliton that appears in solutions in the oscillatory modes and the Rayleigh parameter µ corresponds to the system stiffness. If n = 6 the solutions of equation (2) relate to modes of the hexapod gaits from Fig. 3 where modes a) and b) are doubled for forward and backward walk. In all the oscillatory and non-oscillatory stable solutions the ring is rotating which means that besides the soliton we can observe continuous growth of the output signal. It can be reduced by the last element of the equation and its parameter. This system has been in detail examined in Ref. 4, 5, 6, 7, 8 where the analog representation of the Toda-Rayleigh lattice was built. In this paper it is implemented in the digital system by numerical integration methods. Figure 4. Toda-Rayleigh lattice (ring). 3. Simulated robot motion The simulation model of the whole robot has been built. It consists of the servo motor models identified during the real-time experiments. Each leg is controlled by solving backward and forward kinematics. Instead of setting and reading the natural coordinates of the legs, the Cartesian coordinates of the legs tips are used. The X coordinate corresponds to the front or back motion of the robot. The Z coordinate corresponds to the up or down motion of the robot. The Y coordinate corresponds to the right or left motion of the robot. Figure 5 presents the structure of the leg model. Figure 5. The structure of the one leg model (backward kinematics, three servo motors models and forward kinematics) of the simulation software. The whole robot model has six identical legs. The Toda-Rayleigh lattice model involves the control system of the robot. Every node of the TodaRayleigh lattice is responsible for the generation of a control signal for one robot leg. Figure 6 shows the structure of the Toda-Rayleigh lattice. Figure 6. The Toda-Rayleigh lattice model structure of the simulation software. The solution at every Toda-Rayleigh lattice node is equal to the corresponding leg X coordinate signal. The Y coordinate of the leg is set to a given constant value. It yields that leg tips move only back and forth. The Z coordinate is equal to the first derivative of the X position (this signal is also calculated during numerical integration of the Toda-Rayleigh lattice). These control signals are shown in Figure 7. Figure 7. One leg control signals evaluated from Toda-Rayleigh lattice. The generated control signals invoke the rotary movement of the leg tip in the XZ plane. The trajectory of the expected motion is shown in Figure 8. Figure 8. The trajectory diagram of the expected robot tip motion. Figures 9, 10 and 11 present the simulated leg tip trajectory on the XZ plane in the three possible modes of the Toda-Rayleigh lattice. In these figures one can observe motion disruption resulting from the dynamics of the robot leg. The shape of the generated trajectory varies depending on the Toda-Rayleigh mode. All of these trajectories result in progressive motion of the whole robot. The motion disruption caused by the leg dynamics is smaller than 1 mm. The Toda-Rayleigh lattice is an innovative approach to generate different hexapod gaits and to allow a smooth change of one gait to another. For each gait differences in the cyclical movement of the leg are clearly visible in Figures 9, 10 and 11. Let us note that the expected shape of the orbital motion is an ellipse (see Figure 8). The orbits obtained for the three gaits are very fuzzy ellipses. The gait generator far only shown in the simulation if it is implemented in a real robot it will meet the needs of a real robot walking. As with horses, which can walk, trot and gallop, we recognize three insect gaits: metachronal, caterpillar and tripod. One has to remember that insects do not play free gait. It means that gaits change dynamically only according to the environment. If an obstacle is on the insect path then the leg can be shifted or becomes shorter. It ensures smoothness of the insect motion. Such a behavior of a vivid insect has to be transferred to a walking robot and it becomes a real challenge for the robot design. In the light of the above comment we are justified to consider only a plane surface without any imperfections in the experiments. This assumption will be removed in future studies. Figure 9. Leg tip trajectory controlled by Toda-Rayleigh lattice in its first mode ­ the metachronal gait. Figure 10. Leg tip trajectory controlled by Toda-Rayleigh lattice in its second mode ­ the caterpillar gait. Figure 11. Leg tip trajectory controlled by Toda-Rayleigh lattice in its third mode ­ the tripod gait. 4. Conlusions One can ask a provocative question. Is it not an insult to the human mind that the handmade hexapod and the designed gait algorithm invented by watching nature, is less resistant to interference with environment and disturbances than the walking insect? Certainly not, it just reminds one how much creativity is still needed and it stimulates to further work. The idea of generation of nonlinear oscillations using the Toda-Rayleigh rings seems to be an interesting but complex approach. As it turns out, the application of this approach fits well to a plain surface, but is not sufficient to walk freely on any surface. Therefore the principle of preserving of the potential energy when the Toda-Rayleigh rings controller is used has to be revised for an arbitrary walking surface. The next step of the planned research is to verify the simulated control for the robot by applying it to the real hexapod system. Particular attention will be given to the robustness of the algorithm. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bio-Algorithms and Med-Systems de Gruyter

HEXAPOD - SIX-LEGGED WALKING ROBOT CONTROLLED WITH TODA-RAYLEIGH LATTICE

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Abstract

Movement of six-legged robot is considered. The robot is built to conduct research on the control of different modes of walking with a particular focus on switching from one mode to another. The robot parameters are identified from real-time control experiments. A number of identical servo motors are responsible for the mobility of the hexapod. Therefore, in addition to the construction of hexapod and its control system and the ability to generate different gaits, it is important to create a simulation model of the hexapod including servo motors. The full state observer for the servo motor is an integral part of this model. Forward and backward kinematics of the robot legs is implemented. The six node Toda-Rayleigh lattice model is created and numerically simulated. The lattice is applied as the robot controller. Three different modes of robot gait are achieved. The algorithm used for the gait mode change is equipped with parameters that are experimentally chosen. KEYWORDS: walking robot, walking algorithm, modelling, simulation, identification 1. Introduction The hexapod walking robot equipped with a construction plate is considered. The robot moves through the movements of six identical legs. Every leg consists of three links driven by the Hitec HS-475HB servo motors. The leg construction is similar to the biological construction of the cockroach leg. Thanks to this fact the analysis of natural gaits of insects will be used during the control process design [1]. The robot is shown in Figure 1. The direct control layer with the preset controller in the feedback loop of the servo motor has been implemented by the servo manufacturer. Only the manufacturer is responsible for the tuning of servo controllers, no one else. The whole system includes of 18 servo motors. Figure 1. Photo of the hexapod, a six-legged walking robot. The simulation model of motion is built in Simulink. It is dedicated to the real-time control environment of the hexapod system [2]. The real environment consists of two layers: an FPGA circuit and a PC computer with the real-time operating system [3]. Thanks to six legs we can easily design walking algorithms such that the whole construction will be stable in every phase of walking (contrary to the four-legged and the two-legged robots). The walking algorithms have a very simple form. Usually they correspond to synchronously repetitive actions. In fact, these three algorithms correspond to three different gait modes. Figure 3 shows the way how the legs are moved in each gait mode. Figure 2. The robot leg. Figure 3. Scheme presenting three basic modes of the six-legged insect walk (L stands for left legs and R stands for right legs). In the case of the six-legged insects we consider three characteristic ways of walking: the insect moves one leg in a short period of time, the other five legs support the insect body the insect moves two legs in a short period of time, the other four legs support the insect body the insect moves three legs in a short period of time, the other three legs support the insect body In all simulations and experiments the plane walking surface without any imperfections is considered. The full cycle consists of the following modes: a) six stages, b) three stages and c) only two stages. The analysis of the insect motion helps to develop walking algorithms for the hexapod. One can conclude: at the same velocity of the single leg motion the robot can achieve three different velocities by changing the walk mode the robot can move forward or backward due to the applied sign of the control signal the higher number of legs raised, the faster the robot can move but the legs on which the robot is leaning are more burdened when a leg (or legs) are moving forward, the supporting legs have to move backward with the velocity given by the formula vb = v f n 6n (1) where v f is the velocity of the legs moving forward and n is the number of the legs moving forward. 2. Toda-Rayleigh Lattice (Ring) Cyclic behavior is well known in various scientific and engineering disciplines: in medicine, biology, biochemistry, robotics, etc. It is also important that individual elements of a system are identical and identically related. Therefore we can create spatial-temporal lattice (ring) models. For example, chemical compounds having atoms arrange in rings or closed-chain structures. Particularly, in the robotics discipline, for multi-leg walking robots different types of gait can be represented as the formation and propagation of waves modeled by an evolutionary ring, i.e. a lattice system. The TodaRayleigh lattice is used as a high level control algorithm. The dynamics of the system based on the lattice is described by equation (2). It consists of two main parts: the dissipative Toda lattice and Rayleigh-like energy pumping. If we imagine a chain of point masses coupled by Toda identical springs with dissipative effects then it becomes obvious that decaying nonlinear oscillations have to be stabilized by Rayleigh-type providing or taking away energy. It is interesting that the lattice-ring composed of six non-harmonically interacting devices permits to mimic standard hexapod gaits. The solutions of equation (2) are n - 1 limited cycles, so-called oscillatory modes and two non-oscillatory, rotary modes. The current stable state depends on external parameter pext 2 n = 0 e y yn1 yn yn yn+1 + f ( y 2 pext ) R yn f(x) = ( x 2 ) x (2) Equation (2) describes one node of the lattice (ring). Six identical nodes are connected as it is shown in Figure 4. It describes how the energy is passed over the ring. The parameter 0 is responsible for the frequency of the soliton that appears in solutions in the oscillatory modes and the Rayleigh parameter µ corresponds to the system stiffness. If n = 6 the solutions of equation (2) relate to modes of the hexapod gaits from Fig. 3 where modes a) and b) are doubled for forward and backward walk. In all the oscillatory and non-oscillatory stable solutions the ring is rotating which means that besides the soliton we can observe continuous growth of the output signal. It can be reduced by the last element of the equation and its parameter. This system has been in detail examined in Ref. 4, 5, 6, 7, 8 where the analog representation of the Toda-Rayleigh lattice was built. In this paper it is implemented in the digital system by numerical integration methods. Figure 4. Toda-Rayleigh lattice (ring). 3. Simulated robot motion The simulation model of the whole robot has been built. It consists of the servo motor models identified during the real-time experiments. Each leg is controlled by solving backward and forward kinematics. Instead of setting and reading the natural coordinates of the legs, the Cartesian coordinates of the legs tips are used. The X coordinate corresponds to the front or back motion of the robot. The Z coordinate corresponds to the up or down motion of the robot. The Y coordinate corresponds to the right or left motion of the robot. Figure 5 presents the structure of the leg model. Figure 5. The structure of the one leg model (backward kinematics, three servo motors models and forward kinematics) of the simulation software. The whole robot model has six identical legs. The Toda-Rayleigh lattice model involves the control system of the robot. Every node of the TodaRayleigh lattice is responsible for the generation of a control signal for one robot leg. Figure 6 shows the structure of the Toda-Rayleigh lattice. Figure 6. The Toda-Rayleigh lattice model structure of the simulation software. The solution at every Toda-Rayleigh lattice node is equal to the corresponding leg X coordinate signal. The Y coordinate of the leg is set to a given constant value. It yields that leg tips move only back and forth. The Z coordinate is equal to the first derivative of the X position (this signal is also calculated during numerical integration of the Toda-Rayleigh lattice). These control signals are shown in Figure 7. Figure 7. One leg control signals evaluated from Toda-Rayleigh lattice. The generated control signals invoke the rotary movement of the leg tip in the XZ plane. The trajectory of the expected motion is shown in Figure 8. Figure 8. The trajectory diagram of the expected robot tip motion. Figures 9, 10 and 11 present the simulated leg tip trajectory on the XZ plane in the three possible modes of the Toda-Rayleigh lattice. In these figures one can observe motion disruption resulting from the dynamics of the robot leg. The shape of the generated trajectory varies depending on the Toda-Rayleigh mode. All of these trajectories result in progressive motion of the whole robot. The motion disruption caused by the leg dynamics is smaller than 1 mm. The Toda-Rayleigh lattice is an innovative approach to generate different hexapod gaits and to allow a smooth change of one gait to another. For each gait differences in the cyclical movement of the leg are clearly visible in Figures 9, 10 and 11. Let us note that the expected shape of the orbital motion is an ellipse (see Figure 8). The orbits obtained for the three gaits are very fuzzy ellipses. The gait generator far only shown in the simulation if it is implemented in a real robot it will meet the needs of a real robot walking. As with horses, which can walk, trot and gallop, we recognize three insect gaits: metachronal, caterpillar and tripod. One has to remember that insects do not play free gait. It means that gaits change dynamically only according to the environment. If an obstacle is on the insect path then the leg can be shifted or becomes shorter. It ensures smoothness of the insect motion. Such a behavior of a vivid insect has to be transferred to a walking robot and it becomes a real challenge for the robot design. In the light of the above comment we are justified to consider only a plane surface without any imperfections in the experiments. This assumption will be removed in future studies. Figure 9. Leg tip trajectory controlled by Toda-Rayleigh lattice in its first mode ­ the metachronal gait. Figure 10. Leg tip trajectory controlled by Toda-Rayleigh lattice in its second mode ­ the caterpillar gait. Figure 11. Leg tip trajectory controlled by Toda-Rayleigh lattice in its third mode ­ the tripod gait. 4. Conlusions One can ask a provocative question. Is it not an insult to the human mind that the handmade hexapod and the designed gait algorithm invented by watching nature, is less resistant to interference with environment and disturbances than the walking insect? Certainly not, it just reminds one how much creativity is still needed and it stimulates to further work. The idea of generation of nonlinear oscillations using the Toda-Rayleigh rings seems to be an interesting but complex approach. As it turns out, the application of this approach fits well to a plain surface, but is not sufficient to walk freely on any surface. Therefore the principle of preserving of the potential energy when the Toda-Rayleigh rings controller is used has to be revised for an arbitrary walking surface. The next step of the planned research is to verify the simulated control for the robot by applying it to the real hexapod system. Particular attention will be given to the robustness of the algorithm.

Journal

Bio-Algorithms and Med-Systems – de Gruyter

Published: Jan 1, 2012

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