Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> A set of van der Pol oscillators is arranged in a network in which each oscillator is coupled to each other oscillator. Through the selection of coupling coefficients, the network is made to appear as a ring and as a chain of coupled oscillators. Each oscillator is provided with amplitude, frequency, and offset parameters which have analytically indeterminable effects on the output waves. These systems are simulated on the digital computer in order to study the amplitude, frequency, offset, and phase relationships of the waves versus parameter changes. Based on the simulations, systems of coupled oscillators are configured so that they exhibit stable patterns of signals which can be used to model the central pattern generator (CPG) of living organisms. Using a simple biped as an example locomotory system, the CPG model generates control signals for simulated walking and jumping maneuvers. It is shown that with parameter adjustments, as guided by the simulations, the model can be made to generate kinematic trajectories which closely resemble those for the human walking gait. Further-more, minor tuning of these parameters along with some algebraic sign changes of coupling coefficients can effect a transition in the trajectories to those of a two-legged hopping gait. The generalized CPG model is shown to be versatile enough that it can also generate various n-legged gaits and spinal undulatory motions, as in the swimming motions of a fish. <s> BIB001 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> We present a neural network model inspired from both behavioral and neurophysiological data on insect locomotion. The model consists of a central rhythmic pattern generator and a sensory motor network. We show that it exhibits various behavioral properties observed in several insect species: it produces a continuum of stable gaits ranging from metachronal to tripod; the changes in the duration of protraction retraction phases for various walking speeds follow closely the behavorial data; the network is insensitive to the initial position of the legs, in that it can achieve rapidly a coherent phase relationship regardless of the initial conditions. In addition, this transient property also extends to situations where rapid changes occur in the walking speed; the network can reorganize its phase rapidly during fast accelerations as well as fast decelerations. <s> BIB002 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> The networks of the brainstem and spinal cord that co-ordinate locomotion and body orientation in lamprey are described. The cycle-to-cycle pattern generation of these networks is produced by interacting glutamatergic and glycinergic neurones, with NMDA receptor-channels playing an important role at lower rates of locomotion. The fine tuning of the networks produced by 5-HT, dopamine and GABA systems involves a modulation of Ca2+-dependent K+ channels, high- and low-threshold voltage-activated Ca2+ channels and presynaptic inhibitory mechanisms. Mathematical modelling has been used to explore the capacity of these biological networks. The vestibular control of the body orientation during swimming is exerted via reticulospinal neurones located in different reticular nuclei. These neurones become activated maximally at different angles of tilt. <s> BIB003 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> How do humans and other animals accomplish coordinated movements? How are novel combinations of limb joints rapidly assembled into new behavioral units that move together in in-phase or anti-phase movement patterns during complex movement tasks? A neural central pattern generator (CPG) model simulates data from human bimanual coordination tasks. As in the data, anti-phase oscillations at low frequencies switch to in-phase oscillations at high frequencies, in-phase oscillations occur at both low and high frequencies, phase fluctuations occur at the anti-phase in-phase transition, a “seagull effect” of larger errors occurs at intermediate phases, and oscillations slip toward in-phase and anti-phase when driven at intermediate phases. These oscillations and bifurcations are emergent properties of the CPG model in response to volitional inputs. The CPG model is a version of the Ellias-Grossberg oscillator. Its neurons obey Hodgkin-Huxley type equations whose excitatory signals operate on a faster time scale than their inhibitory signals in a recurrent on-center off-surround anatomy. When an equal command or GO signal activates both model channels, the model CPG can generate both in-phase and anti-phase oscillations at different GO amplitudes. Phase transitions from either in-phase to anti-phase oscillations, or from anti-phase to in-phase oscillations, can occur in different parameter ranges, as the GO signal increases. <s> BIB004 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Recently, the field of humanoid robotics attracts more and more interest and the research on humanoid locomotion based on central pattern generators (CPG) reveals many challenging aspects. This paper describes the design of CPG for stable humanoid bipedal locomotion using an evolutionary approach. In this research, each joint of the humanoid is driven by a neuron that consists of two coupled neural oscillators, and corresponding joint's neurons are connected by strength weight. To achieve natural and robust walking pattern, an evolutionary-based multi-objective optimization algorithm is used to solve the weight optimization problem. The fitness functions are formulated based on zero moment point (ZMP), global attitude of the robot and the walking speed. In the algorithms, real value coding and tournament selection are applied, the crossover and mutation operators are chosen as heuristic crossover and boundary mutation respectively. Following evolving, the robot is able to walking in the given environment and a simulation shows the result. <s> BIB005 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> We have been trying to induce a quadruped robot to walk with medium walking speed on irregular terrain based on biological concepts. We propose the necessary conditions for stable dynamic walking on irregular terrain in general, and we design the mechanical system and the neural system by comparing biological concepts with those necessary conditions described in physical terms. A PD controller at the joints can construct the virtual spring-damper system as the visco-elasticity model of a muscle. The neural system model consists of a central pattern generator (CPG) and reflexes. A CPG receives sensory input and changes the period of its own active phase. The desired angle and P-gain of each joint in the virtual spring-damper system is switched based on the phase signal of the CPG. CPGs, the motion of the virtual spring-damper system of each leg and the rolling motion of the body are mutually entrained through the rolling motion feedback to CPGs, and can generate adaptive walking. We report on our experimen... <s> BIB006 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> This paper proposes a neuromorphic analog CMOS controller for interlimb coordination in quadruped locomotion. Animal locomotion, such as walking, running, swimming, and flying, is based on periodic rhythmic movements. These rhythmic movements are driven by the biological neural network, called the central pattern generator (CPG). In recent years, many researchers have applied CPG to locomotion controllers in robotics. However, most of these have been developed with digital processors and, thus, have several problems, such as high power consumption. In order to overcome such problems, a CPG controller with analog CMOS circuit is proposed. Since the CMOS transistors in the circuit operate in their subthreshold region and under low supply voltage, the controller can reduce power consumption. Moreover, low-cost production and miniaturization of controllers are expected. We have shown through computer simulation, such circuit has the capability to generate several periodic rhythmic patterns and transitions between their patterns promptly. <s> BIB007 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Autonomic oscillatory activities exist in almost every living thing and most of them are produced by rhythmic activities of the corresponding neural systems (locomotion, respiration, heart beat, etc.). This paper mathematically discusses sustained oscillations generated by mutual inhibition of the neurons which are represented by a continuous-variable model with a kind of fatigue or adaptation effect. If the neural network has no stable stationary state for constant input stimuli, it will generate and sustain some oscillation for any initial state and for any disturbance. Some sufficient conditions for that are given to three types of neural networks: lateral inhibition networks of linearly arrayed neurons, symmetric inhibition networks and cyclic inhibition networks. The result suggests that the adaptation of the neurons plays a very important role for the appearance of the oscillations. Some computer simulations of rhythmic activities are also presented for cyclic inhibition networks consisting of a few neurons. <s> BIB008 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> In this paper, we develop a simple bipedal locomotion algorithm based on biological concepts. The algorithm utilized a central pattern generator (CPG) composed of four coupled neural oscillators (NO) to generate control signal for the bipedal robot. Feedbacks from the robot dynamics and the environment are used to update the CPG online. Our algorithm is then tested on a seven link model of bipedal robot. Simulation results suggest the proposed CPG can generate a smooth and continuous walking pattern for the robot. <s> BIB009 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> This paper presents simulation and experimental results of dynamic walking and running of a bipedal robot. We proposed the hybrid central pattern generator (H-CPG) method to realize adaptive dynamic motions including stepping and jumping. This method basically consisted of CPG models, but was added the force control system that controlled the acting force from a leg to a floor in the vertical and the horizontal directions separately. At this moment, 2D walking and running was realized on level ground and a slope with a speed of 1.6 m/s. <s> BIB010 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Abstract In this paper, we investigated the effects of the friction condition on walking pattern and energy efficiency, and based on the results, we proposed two new “slip-adaptive” strategies for generating a slip-adaptive walk. The first strategy for a slip-adaptive walk uses a slip reflex via a Central Pattern Generator (CPG) to change the walking pattern. The second strategy for a walk uses a force control to immediately compensate a slip. Using these strategies, a walk, which is adaptive to varying friction conditions and slips, becomes possible. The validity of the proposed method is confirmed through simulation and experimentation. <s> BIB011 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> In this paper, we present our design and experiments on a planar biped robot under the control of a pure sensor-driven controller. This design has some special mechanical features, for example small curved feet allowing rolling action and a properly positioned center of mass, that facilitate fast walking through exploitation of the robot's natural dynamics. Our sensor-driven controller is built with biologically inspired sensor- and motor-neuron models, and does not employ any kind of position or trajectory tracking control algorithm. Instead, it allows our biped robot to exploit its own natural dynamics during critical stages of its walking gait cycle. Due to the interaction between the sensor-driven neuronal controller and the properly designed mechanics of the robot, the biped robot can realize stable dynamic walking gaits in a large domain of the neuronal parameters. In addition, this structure allows the use of a policy gradient reinforcement learning algorithm to tune the parameters of the sensor-driven controller in real-time, during walking. This way RunBot can reach a relative speed of 3.5 leg lengths per second after only a few minutes of online learning, which is faster than that of any other biped robot, and is also comparable to the fastest relative speed of human walking. <s> BIB012 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> We present a system of coupled nonlinear oscillators to be used as programmable central pattern generators, and apply it to control the locomotion of a humanoid robot. Central pattern generators are biological neural networks that can produce coordinated multidimensional rhythmic signals, under the control of simple input signals. They are found both in vertebrate and invertebrate animals for the control of locomotion. In this article, we present a novel system composed of coupled adaptive nonlinear oscillators that can learn arbitrary rhythmic signals in a supervised learning framework. Using adaptive rules implemented as differential equations, parameters such as intrinsic frequencies, amplitudes, and coupling weights are automatically adjusted to replicate a teaching signal. Once the teaching signal is removed, the trajectories remain embedded as the limit cycle of the dynamical system. An interesting aspect of this approach is that the learning is completely embedded into the dynamical system, and does not require external optimization algorithms. We use our system to encapsulate rhythmic trajectories for biped locomotion with a simulated humanoid robot, and demonstrate how it can be used to do online trajectory generation. The system can modulate the speed of locomotion, and even allow the reversal of direction (i.e. walking backwards). The integration of sensory feedback allows the online modulation of trajectories such as to increase the basin of stability of the gaits, and therefore the range of speeds that can be produced <s> BIB013 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> In limbed animals, spinal neural circuits responsible for controlling muscular activities during walking are called central pattern generators (CPG). CPG networks display oscillatory activities that actuates individual or groups of muscles in a coordinated fashion so that the limbs of the animal are flexed and extended at the appropriate time and with the required velocity for the animal to efficiently traverse various types of terrain, and to recover from environmental perturbation. Typically, the CPG networks are constructed with many neurons, each of which has a number of control parameters. As the number of muscles increases, it is often impossible to manually, albeit intelligently, select the network parameters for a particular movement. Furthermore, it is virtually impossible to reconfigure the parameters on-line. This paper describes how genetic algorithms (GA) can be used for on-line (re)configuring of CPG networks for a bipedal robot. We show that the neuron parameters and connection weights/network topology of a canonical walking network can be reconfigured within a few of generations of the GA. The networks, constructed with integrate-and-fire-with-adaptation (IFA) neurons, are implemented with a microcontroller and can be reconfigured to vary walking speed from 0.5Hz to 3.5Hz. The phase relationship between the hips and knees can be arbitrarily set (to within 1 degree) and prescribed complex joint angle profiles are realized. This is a powerful approach to generating complex muscle synergies for robots with multiple joints and distributed actuators. <s> BIB014 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Central pattern generator (CPG) is a neural circuit which governs rhythmic activities of animal. Synthesis of CPG plays an important role in engineering such as locomotion control. Up to now, CPGs are mainly derived based on biological principles, with their parameters heuristically tuned. In this paper, a class of piecewise affine systems capable of exhibiting CPG-like property of stable limit cycles is presented. A proposed synthesis method for CPGs is also included. <s> BIB015 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Biped robot with heterogeneous legs (BRHL) is a novel robot model, which consists of an artificial leg and an intelligent bionic leg. The artificial leg is used to simulate the amputeepsilas healthy leg and the bionic leg works as the intelligent artificial limb. To describe the present gait of the healthy leg and make intelligent bionic leg follow the walking of artificial leg in all phases is the target of BRHLpsilas research. So gait planning of artificial leg is the emphasis of BRHLpsilas research. This paper uses the model of central pattern generator (CPG) in the research of artificial legpsilas gait planning from the point of biology. To obtain natural and robust walking pattern, genetic algorithm is used to optimize parameters of the CPG network model and the fitness function is formulated based on zero moment point (ZMP). Simulation results testify the feasibility of this method. <s> BIB016 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> In this paper we describe a learning framework for a central pattern generator (CPG)-based biped locomotion controller using a policy gradient method. Our goals in this study are to achieve CPG-based biped walking with a 3D hardware humanoid and to develop an efficient learning algorithm with CPG by reducing the dimensionality of the state space used for learning. We demonstrate that an appropriate feedback controller can be acquired within a few thousand trials by numerical simulations and the controller obtained in numerical simulation achieves stable walking with a physical robot in the real world. Numerical simulations and hardware experiments evaluate the walking velocity and stability. The results suggest that the learning algorithm is capable of adapting to environmental changes. Furthermore, we present an online learning scheme with an initial policy for a hardware robot to improve the controller within 200 iterations. <s> BIB017 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> A control problem is studied for generating the locomotion pattern of a semi-looper type robot by applying central pattern generators (CPGs), in which such a robot can realize two-rhythm motion and green caterpillar locomotion depending on the condition of environment. After deriving the dynamical model with two links and one actuator, the simulation of the robot is conducted using a CPG consisting of one van der Pol (VDP) oscillator. A CPG network composed of two VDP oscillators is further constructed to realize four-rhythm motion with three links and two actuators. <s> BIB018 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> The generation of trajectories for a biped robot is a problem which has been largely studied for several years, and many satisfying offline solutions exist for steady-state walking in absence of disturbances. The question is a little more complex when the generation of the desired trajectories of joints or links has to be achieved or adapted online, i.e. in real time, for example when it is wished to strongly synchronize these trajectories with an external motion. This is precisely the problem addressed in this paper. Indeed, we consider the case where the ''master'' motion is measured by a position sensor embedded on a human leg. We propose a method to synchronize the motion of a robot or of other device with respect to the output signal of the sensor. The main goal is to estimate as accurately as possible the current phase along the gait cycle. We use for that purpose a model based on a nonlinear oscillator, which we associate an observer. Introducing the sensor output in the observer allows us to compute the oscillator phase and to generate a synchronized multilinks trajectory, at a very low computational cost. The paper also presents evaluation results in terms of robustness against parameter estimation errors and velocity changes in the input. <s> BIB019 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Central pattern generators (CPGs) are becoming a popular model for the control of locomotion of legged robots. Biological CPGs are neural networks responsible for the generation of rhythmic movements, especially locomotion. In robotics, a systematic way of designing such CPGs as artificial neural networks or systems of coupled oscillators with sensory feedback inclusion is still missing. In this contribution, we present a way of designing CPGs with coupled oscillators in which we can independently control the ascending and descending phases of the oscillations (i.e. the swing and stance phases of the limbs). Using insights from dynamical system theory, we construct generic networks of oscillators able to generate several gaits under simple parameter changes. Then we introduce a systematic way of adding sensory feedback from touch sensors in the CPG such that the controller is strongly coupled with the mechanical system it controls. Finally we control three different simulated robots (iCub, Aibo and Ghostdog) using the same controller to show the effectiveness of the approach. Our simulations prove the importance of independent control of swing and stance duration. The strong mutual coupling between the CPG and the robot allows for more robust locomotion, even under non precise parameters and non-flat environment. <s> BIB020 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> We present a new quadruped robot, ldquoCheetahrdquo, featuring three-segment pantographic legs with passive compliant knee joints. Each leg has two degrees of freedom - knee and hip joint can be actuated using proximal mounted RC servo motors, force transmission to the knee is achieved by means of a bowden cable mechanism. Simple electronics to command the actuators from a desktop computer have been designed in order to test the robot. A Central Pattern Generator (CPG) network has been implemented to generate different gaits. A parameter space search was performed and tested on the robot to optimize forward velocity. <s> BIB021 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> Central pattern generators (CPGs) inspired from neural system of animal are widely used for the control of locomotion in robots. The objective of this paper is to modeling a CPG-network formed by a set of mutually coupled Van Der Pol (VDP) oscillators to generating rhythmic movement patterns for multi-joint robot. Firstly, a VDP-CPG network is made up by four coupled VDP oscillators, which can produce multiple phase-locked oscillation patterns that correspond to the four basic quadrupedal gaits. Then the gait transitions between the different gaits are generated by altering the internal oscillator parameters. At last, we use VDP-CPG network to produce the joints trajectories for AIBO to control the locomotion. The simulation and experimental results demonstrate that the designed VDP-CPG network is effective to control the quadruped locomotion. <s> BIB022 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> The objective of the work presented here is to develop a low cost active above knee prosthetic device exploiting bipedal robotics technology which will work utilizing the available biological motor control circuit properly integrated with a Central Pattern Generator (CPG) based control scheme. The approach is completely different from the existing Active Prosthetic devices, designed primarily as standalone systems utilizing multiple sensors and embedded rigid control schemes. In this research, first we designed a fuzzy logic based methodology for offering suitable gait pattern for an amputee, followed by formulating a suitable algorithm for designing a CPG, based on Rayleigh's oscillator. An indigenous probe, Humanoid Gait Oscillator Detector (HGOD) has been designed for capturing gait patterns from various individuals of different height, weight and age. These data are used to design a Fuzzy inference system which generates most suitable gait pattern for an amputee. The output of the Fuzzy inference system is used for designing a CPG best suitable for the amputee. We then developed a CPG based control scheme for calculating the damping profile in real time for maneuvering a prosthetic device called AMAL (Adaptive Modular Active Leg). Also a number of simulation results are presented which show the stable behavior of knee and hip angles and determine the stable limit cycles of the network. <s> BIB023 </s> Intelligent Approaches in Locomotion - A Review <s> Systems of differential equations <s> In this paper, we propose a new CPG model and hebbian learning rule for the CPG. The output of the proposed CPG is determined by only phase differences of synchronization of component oscillators. A phase synchronization can be regarded as an adaptive behavior to an environment, so the CPG has an adaptavility despite of only simple connections to the environment. We also propose a motion learning rule for the proposed CPG. Since the rule is described by only simple signal processing, so that it can be realized by electric circuits easily, it will be able to be used efficiently for high degree of freedom robots. <s> BIB024
Analysis of biological CPGs has identified oscillators made from pairs of mutually inhibiting neurons BIB003 . These can be modelled or approximated by systems of differential equations. The solutions of these systems, and variations on this theme, range from simple sinusoidal forms to more complex shapes. Matsuoka oscillators are commonly used for CPGs BIB016 , BIB009 , BIB010 , BIB017 ,  , BIB006 , BIB011 , BIB005 -  , BIB008 , and one reason for their popularity is that their dynamics, including limit cycle behaviour, are well known BIB008 . Matsuoka oscillators are capable of different gaits, such as the walk, trot and pace quadruped gaits, by specifying different phase relationships BIB006 . They are also capable of smooth gait transitions, which were dramatically demonstrated in  where a 2D biped switched to climb on all fours up the slope. Using sensory information, Matsuoka CPGs are able to control walking not just on a flat surface but on an irregular BIB006 or slippery surfaces BIB011 . Van der Pol oscillators (VDP) are also popular for CPGs BIB018 , BIB001 , BIB022 , BIB019 . The benefits of these oscillators are that they have stable limit cycles and relatively interpretable coefficients. Frequency, amplitude and shape coefficients can be identified, although they are not completely independent. Different gaits are possible with VDP oscillators, including walk, trot, pace and bound quadruped gaits BIB001 , walking and forward jumping biped gaits BIB022 . In both situations, the ability to transition between gaits was developed through simple changes to the parameters of the oscillators. Rayleigh oscillators, which are similar to VDP oscillators, were used in BIB023 to construct a CPG for an active prosthetic leg. Reduced sensory requirements were highlighted as an advantage over conventional prosthetic systems. Similar to the Matsuoka oscillators in form, Amari-Hopfield oscillators are also used in CPGs BIB007 , BIB012 . In BIB012 they generated biped locomotion that was resilient to perturbations as a result of the limit cycle properties of the oscillator. In BIB007 , they were able to produce walk, trot and gallop quadruped gaits, with prompt gait transitions. Hopf oscillators were used in BIB020 , BIB021 , BIB024 for quadruped locomotion. In BIB020 , they were chosen for their stable limit cycle (even when the frequency changes), and for their independence of cycle shape and frequency. In BIB013 , a series of Hopf oscillators was investigated, that included feedback terms allowing learning of an input trajectory, with each sub-oscillator matching a partial of the input. A CPG using these combined oscillators was trained with a reference bipedal locomotion pattern, thus converting a reference trajectory into a system with limit cycle properties, so that it became resilient to perturbation. Spiking Integrate and Fire with Adaption neurons were used in BIB014 for biped locomotion. The CPG had a hierarchical structure of hip timing, knee timing, and finally output patterning. The core oscillator consisted of a neuron pair that controlled the hip timing, which in turn coupled to a pair for the knee timing, therefore establishing the gait characteristics. Finally, the output of the timing neurons was sent to sets of parallel motor neurons that had independent configurable firing characteristics. They therefore allowed production of complex patterns that were driven by the periodicity of the timing neurons. This three sub-system design allowed separate configuration of walking frequency, gait and joint angle profile, by adjusting the parameters of the hip oscillator, the hip-knee coupling, and the motor neurons respectively. Generalized CPGs were developed in BIB015 using piecewise affine systems. Analysis was given that allowed amplitude and frequency to be independently specified. The paper cited flexibility, and linear relationships between input frequency and amplitude controls and the output frequency and amplitude, as benefits of this approach. Another type of biologically inspired oscillators is the Ellias shunting oscillator, which was used in BIB002 , BIB004 .