Derivation and Application of a Conserved Orbital Energy for the Inverted Pendulum Bipedal Walking Model

We present an analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking. Using this model, we derive expressions for a conserved quantity, the “Orbital Energy”, given a smooth Center of Mass trajectory. Given a closed form Center of Mass Trajectory, the equation for the Orbital Energy is a closed form expression except for an integral term, which we show to be the first moment of area under the Center of Mass path. Hence, given a Center of Mass trajectory, it is straightforward and computationally simple to compute phase portraits for the system. In fact, for many classes of trajectories, such as those in which height is a polynomial function of Center of Mass horizontal displacement, the Orbital Energy can be solved in closed form. Given expressions for the Orbital Energy, we can compute where the foot should be placed or how the Center of Mass trajectory should be modified in order to achieve a desired velocity on the next step. We demonstrate our results using a planar biped simulation with light legs and point mass body. We parameterize the Center of Mass trajectory with a fifth order polynomial function. We demonstrate how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity

Derivation and Application of a Conserved Orbital Energy for the Inverted Pendulum Bipedal Walking Model

Abstract—We present an analysis of a point mass, point foot, planar inverted pendulum model for bipedal walking. Using this model, we derive expressions for a conserved quantity, the “Orbital Energy”, given a smooth Center of Mass trajectory. Given a closed form Center of Mass Trajectory, the equation for the Orbital Energy is a closed form expression except for an integral term, which we show to be the first moment of area under the Center of Mass path. Hence, given a Center of Mass trajectory, it is straightforward and computationally simple to compute phase portraits for the system. In fact, for many classes of trajectories, such as those in which height is a polynomial function of Center of Mass horizontal displacement, the Orbital Energy can be solved in closed form. Given expressions for the Orbital Energy, we can compute where the foot should be placed or how the Center of Mass trajectory should be modified in order to achieve a desired velocity on the next step. We demonstrate our results using a planar biped simulation with light legs and point mass body. We parameterize the Center of Mass trajectory with a fifth order polynomial function. We demonstrate how the parameters of this polynomial and step length can be changed in order to achieve a desired next step velocity. I

Intelligent model-based control of complex three-link mechanisms

The aim of this study is to understand the complexity and control challenges of the locomotion of a three-link mechanism of a robot system. In order to do this a three-link robot gymnast (Robogymnast) has been built in Cardiff University. The Robogymnast is composed of three links (one arm, one torso, one leg) and is powered by two geared DC motors. Currently the robot has three potentiometers to measure the relative angles between adjacent links and only one tachometer to measure the relative angular position of the first link. A mathematical model for the robot is derived using Lagrange equations. Since the model is inherently nonlinear and multivariate, it presents more challenges when modelling the Robogymnast and dealing with control motion problems. The proposed approach for dealing with the design of the control system is based on a discrete-time linear model around the upright position of the Robogymnast. To study the swinging motion of the Robogymnast, a new technique is proposed to manipulate the frequency and the amplitude of the sinusoidal signals as a means of controlling the motors. Due to the many combinations of the frequency and amplitude, an optimisation method is required to find the optimal set. The Bees Algorithm (BA), a novel swarm-based optimisation technique, is used to enhance the performance of the swinging motion through optimisation of the manipulated parameters of the control actions. The time taken to reach the upright position at its best is 128 seconds. Two different control methods are adopted to study the balancing/stablising of the Robogymnast in both the downward and upright configurations. The first is the optimal control algorithm using the Linear Quadratic Regulator (LQR) technique with integrators to help achieve and maintain the set of reference trajectories. The second is a combination of Local Control (LC) and LQR. Each controller is implemented via reduced order state observer to estimate the unmeasured states in terms of their relative angular velocities. From the identified data in the relative angular positions of the upright balancing control, it is reported that the maximum amplitude of the deviation in the relative angles on average are approximately 7.5° for the first link and 18° for the second link. It is noted that the third link deviated approximately by 2.5° using only the LQR controller, and no significant deviation when using the LQR with LC. To explore the combination between swinging and balancing motions, a switching mechanism between swinging and balancing algorithm is proposed. This is achieved by dividing the controller into three stages. The first stage is the swinging control, the next stage is the transition control which is accomplished using the Independent Joint Control (IJC) technique and finally balancing control is achieved by the LQR. The duration time of the transition controller to track the reference trajectory of the Robogymnast at its best is found to be within 0.4 seconds. An external disturbance is applied to each link of the Robogymnast separately in order to study the controller's ability to overcome the disturbance and to study the controller response. The simulation of the Robogymnast and experimental realization of the controllers are implemented using MATLAB® software and the C++ program environment respectively

A Foot Placement Strategy for Robust Bipedal Gait Control

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance

Intelligent System Synthesis for Dynamic Locomotion Behavior in Multi-legged Robots

Robot technology has been implemented in many fields of our life, such as entertainment, security, rescue, rehabilitation, social life, the military, and etc. Multi-legged robot always exist in many fields, therefore it is important to be developed. Motion capabilities of the robot will be a main focus to be developed. Current development or conventional model of motion capabilities have several issues in saturation of development. There are some limitation in dynamic factors such as, locomotion generator, flexibility of motion planning, and smoothness of movement. Therefore, in this research, natural based computation are implemented as the basic model. There are three subsystems to be developed and integrated, (1) locomotion behavior model, (2) stability behavior model, and (3) motion planning model. Since individual people has different walking behavior in each walking direction and walking speed, locomotion behavior learning model of omni-directional bio-inspired locomotion which is generating different walking behavior in different walking provision are required to be developed. Step length in sagital and coronal direction, and degree of turning are considered parameters in walking provision. In proposed omni-directional walking model, interconnection structures composed by 16 neurons where 1 leg is represented by 4 joints and 1 joint is represented by 2 motor neurons. In order to acquire walking behavior in certain walking provision, the interconnection structure is optimized by multi-objectives evolutionary algorithm. For acquiring the diversity of references, several optimized interconnection structures are generated in optimization processes in different walking provisions. Learning models are proposed for solving non-linearity of relationship between walking input and walking output representing the synaptic weight of interconnection structure, where one learning model representing one walking parameter. Furthermore, by using optimized model, walking behavior can be generated with unsealed walking provision. Smooth walking transition with low error of desired walking provision was proved based on several numerical experiments in physical computer simulation. In stability behavior model, neuro-based push recovery controller is applied in multi-legged robot in order to keep the stability with minimum energy required. There are three motion patterns in individual people behavior when it gets external perturbation, those are ankle behavior, hip behavior, and step behavior. We propose a new model of Modular Recurrent Neural Network (MRNN) for performing online learning system in each motion behavior. MRNN consists of several recurrent neural networks (RNNs) working alternatively depending on the condition. MRNN performs online learning process of each motion behavior controller independently. The aim of push recovery controller is to manage the motion behavior controller by minimizing the energy required for responding to the external perturbation. This controller selects the appropriate motion behavior and adjusts the gain that represent the influence of the motion behavior to certain push disturbance based on behavior graphs which is generated by adaptive regression spline. We applied the proposed controller to the humanoid robot that has small footprint in open dynamics engine. Experimental result shows the effectiveness of the push controller stabilizing the external perturbation with minimum energy required. Proposed motion planning model presents a natural mechanism of the human brain for generating a dynamic path planning in 3-D rough terrain. The proposed model not only emphasizes the inner state process of the neuron but also the development process of the neurons in the brain. There are two information transmission processes in this proposed model, the forward transmission activity for constructing the neuron connections to find the possible way and the synaptic pruning activity with backward neuron transmission for finding the best pathway from current position to target position and reducing inefficient neuron with its synaptic connections. In order to respond and avoid the unpredictable obstacle, dynamic path planning is also considered in this proposed model. An integrated system for applying the proposed model in the actual experiments is also presented. In order to confirm the effectiveness of the proposed model, we applied the integrated system in the pathway of a four-legged robot on rough terrain in computer simulation. For analyzing and proving the flexibility of proposed model, unpredictable collision is also performed in those experiments. The model can find the best pathway and facilitate the safe movement of the robot. When the robot found an unpredictable collision, the path planner dynamically changed the pathway. The proposed path planning model is capable to be applied in further advance implementation. In order to implement the motion capabilities in real cases, all subsystem should be integrated into one interconnected motion capabilities model. We applied small quadruped robot equipped with IMU, touch sensor, and dual ultrasonic sensor for performing motion planning in real terrain from starting point to goal point. Before implemented, topological map is generated by Kinect camera. In this implementation, all subsystem were analyzed and performed well and the robot able to stop in the goal point. These implementation proved the effectiveness of the system integration, the motion planning model is able to generate safe path planning, the locomotion model is able to generate flexible movement depending on the walking provision from motion planning model, and the stability model can stabilize the robot on rough terrain. Generally, the proposed model can be expected to bring a great contribution to the motion capabilities development and can be used as alternative model for acquiring the dynamism and efficient model in the future instead of conventional model usage. In the future, the proposed model can be applied into any legged robot as navigation, supporter, or rescue robot in unstable environmental condition. In addition, we will realize a cognitive locomotion that generates multiple gaits depending on the 3 aspects, embodiment, locomotion generator, and cognition model. A dynamic neuro-locomotion integrated with internal and external sensory information for correlating with the environmental condition will be designed.ロボット技術は、エンターテイメント、セキュリティ、救助、リハビリ、社会生活、軍事などの様々な生活分野に実現さている。多脚ロポットは常に多くの分野に存在するため開発することが重要である。ロボットの運動能力が開発の主要となっている。現状の開発されている動作能力は,飽和状態にある。いくつかの動的な要因により、歩行生成器、動作計画の柔軟性、および動作の滑らかさ等に制限がある。そこで、本研究では、基本的なモデルとして自然計算に基づく方法論を実装する、また、本研究では、歩行動作モデル、安定動作モデル、や運動計画モデルからなる3つのサブシステムを開発し統合する。人間は歩行方向と速度に応じて歩行動作が異なるため、異なる歩行軸では異なる歩行動作を生成するという全方位生物的な運動の歩行動作学習モデルが開発には要求される。球欠および制御方向のステップ長や旋回の度合いは,歩行軸のパラメータとして考慮される。提案した全方位歩行モデルでは,1肢につき16個のニューロンによって構成される相互接続構造を4つの関節によって表現する。また、1つの関節は,2個のモータニューロンによって表現する。一定の歩行軸での歩行動作を獲得するために,本研究では,多目的進化アルゴリズムによって最適化を行う。提案手法では、参照点の多様性を獲得するために,異なる歩行軸においていくつかの最適な相互接続構造が生成される。相互接続構造のシナプス重みを表現している歩行入力と出力間の非線形な関係を解くための学習モデルを構築する。本手法では,1つの学習モデルが1つの歩行パラメータで表現され、最適化されたモデルを用いることにより,歩行動作は,スケーリングされていない歩行軸を生成することが可能となる,物理演算シミュレーションを用いた実験により,誤差の少ない歩行軸の滑らかな歩行遷移を本実験では示している。安定動作モデルでは、必要最小限のエネルギーで安定性を維持するため多足歩行ロボットにニューロベースプッシュリカバリ制御器を適用した。外力をを受けたとき,人間の行動には足首の動作・股関節の動作・踏み動作の3つの動作パターンが存在する。本研究では,各運動動作におけるオンライン学習システムを実現するために、モジュラーリカレントニューラルネットワーク(MRNN)を用いた新たな学習モデルを提案する。MRNNは状況に応じて選択される複数のリカレントニューラルネットワーク(RNN)によって構成される。MRNNは各運動動作コントローラのオンライン学習プロセスを独立して実行する。プッシュリカバリ制御器の目的は、外乱に応じてエネルギー最小化を行うことによって運動動作制御器を管理することである。この制御器は適切な運動動作を選択し,適応回帰スプラインにより生成された動作グラフに基づき押し動作に対して最も影響を及ぼす運動動作のゲインの調整を行う。提案した制御器をOpen Dynamics Engine(ODE)上で小さな足の長さを持つヒューマノイドロボットに適用し,必要最小限のエネルギーで外力に対して安定させるプッシュリカバリ制御器の有効性を示している。3次元の不整地における動的な経路計画を生成するために,人間の自然な脳機能に基づいた動作計画手法を提案する。本モデルは、ニューロンの内部状態過程だけでなく、脳内のニューロンの発達過程も重視している。本モデルは二つのアルゴリズムに構成される。1つは、通過可能な道を見つけるために構築される接続的なニューロン活動である順方向伝達活動であり,もう1つは、現在位置から最適経路を見つけるために、シナプス結合を用いて非効率的なニューロンを減少させる逆方向にニューロン伝達を行うシナプスプルーニング活動である。また,予測不可能な衝突を回避するために,動的な経路計画も実行される。さらに、実環境において提案されたモデルを実現するための統合システムも提示される。提案モデルの有効性を検証するために,コンピュータシミュレーション上で、不整地環境の4足歩行ロボットに関するシミュレーション環境を実装した。これらの実験では,予測不能な衝突に関する実験も行った。本モデルは、最適経路を見つけ出しロボットの安全な移動を実現できた。さらに、ロボットが予測できない衝突を検出した場合,経路計画アルゴリズムが経路を動的に変更可能であることを示している。これらのことから、提案された経路計画モデルはさらなる先進的な展開が実現可能であると考えられる。実環境における運動能力を実装するためには、すべてのサブシステムを1つの運動能力モデルに統合する必要がある。そこで本研究では、IMU、タッチセンサ、2つの超音波センサを搭載した小型の4足歩行ロポットを用いた実環境において出発地点から目的地点までの運動計画を行った、本実装では、3次元距離計測センサであるKinecを用い3次元空間の位相構造を生成する。また、本実装では、すべてのサブシステムが分析され、ロボットは目的地点で停止することができた。さらに、安全な経路計画を生成することができたことからシステム統合の有効性が確認できた。また、歩行モデルにより歩行軸に応じた柔軟な動きが生成されることで、この安定性モデルは不整地環撹でもロボットの歩行を安定させることができた。これらのことから、本提案モデルは運動能力への多大な貢献が期待され、ダイナミクスを獲得するための代替モデルとして使用することができ,現在よく使用されているモデルに代わる効率的なモデルとなることが考えられる。今後の課題としては,不安定な環境下におけるナビゲーション・支援・レスキューロボットといった任意の肢の数を持つ多足歩行ロボットへの本提案モデルの適用があげられる。さらに,身体性,歩行生成,認知モデルの3つの観点から複数の歩容を生成する認知的歩行を実現することを考えている。環境と相互作用するためのモデルとして、内界センサと外界センサ情報を統合した動的ニューロ歩行を実現する予定である。首都大学東京, 2018-03-25, 修士（工学）首都大学東