Omnidirectional walking of legged robots with a failed leg

This paper studies omnidirectional walking of a hexapod robot with a locked joint failure by proposing crab gaits and turning gaits. Due to the reduced workspace of a failed leg, fault-tolerant gaits have limitations in their mobility. As for crab gaits, an accessible range of the crab angle is derived for a given configuration of the failed leg. As for turning gaits, the conditions on turning trajectories guaranteeing fault tolerance are derived for spinning gaits and circling gaits. Based on the principles of fault-tolerant gait planning, periodic crab gaits and turning gaits are proposed in which a hexapod robot realizes tripod walking after a locked joint failure, having a reasonable stride length and stability margin. The proposed fault-tolerant gaits are then applied to an obstacle avoidance problem of a hexapod robot with a locked joint failure. The kinematic constraints of fault-tolerant gaits should be considered in planning the robot trajectory.     

Investigation of optimal bipedal walking gaits subject to different energy-based objective functions

Optimal bipedal walking gaits subject to different energy-based objective functions are investigated using a simple planar rigid body model of a bipedal robot with upper body, thighs and shanks. The robot's segments are connected by revolute joints actuated by electric motors. The actuators' torques are generated by a trajectory tracking controller to produce periodic walking gaits. A numerical optimization routine is used to find optimal reference trajectories for average speeds in the range of 0.3 – 2.3 m/s to investigate the influence of different objective functions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Virtual quadruped: Mechanical design,control, simulation,and experimentation

We consider a prototyped walking robot containing a platform and two double-link legs. Thus, it is a five-link mechanism. The front leg models identical motions of the quadruped’s two front legs, and the back leg models identical motions of the quadruped’s two back legs. The legs have passive (uncontrolled) feet that extend in the frontal plane. Because of this the robot is stable in the frontal plane. This robot can be viewed as a “virtual” quadruped. Four DC motors drive the mechanism. Its control system comprises a computer, hardware servo-systems, and power amplifiers. The locomotion of the prototype is planar curvet gait. In the double support our prototype is statically stable and overactuated. In the single support it is an unstable and underactuated system. There is no flight phase. We describe here the scheme of the mechanism, the characteristics of the drives, and the control strategy. The dynamic model of the planar walking is recalled for the double-and single-support phases and for the impact instant. The experiments give results that are close to those of the simulation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 5–28, 2005.     

Energy-optimized bipedal running of a simple humanoid robot

A method to optimize energy efficiency for bipedal running robots is presented. A running model of a simple bipedal robot consisting of five rigid bodies connected by actuated revolute joints is introduced. The actuators' torques are generated by a trajectory tracking controller to produce periodic running gaits. The controller's reference trajectories are parameterized by Bézier polynomials. A numerical optimization is used to employ reference trajectories with optimal energy efficiency for average velocities in the range of 1.5 to 5.5 m/s. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of static optimization criteria for resolving the 3D muscle redundancy problem during human gait

In this paper, different approaches of static optimization for predicting muscle forces during human walking are investigated. In order to better reflect the true mechanics of the human body, a three-dimensional musculoskeletal model of a single leg is developed. The joint moments generated by muscles during walking are computed from inverse dynamics. The muscle force is estimated by different optimization criteria, each satisfying the moment constraints at all joints and the lower and upper muscle force constraints. Several polynomial and non-polynomial criteria frequently used in literature are studied. Then the results obtained from these calculations are compared with each other. This paper provides an overview of the effects of different optimization criteria on the 3D muscle force distribution problem during human walking. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

基于人机耦合的下肢外骨骼动力学分析及仿真

建立了一种包含人机交互力的人体-外骨骼模型，对人体和外骨骼分别采用7连杆的刚体模型进行建模，建立其D-H坐标系，得到人机模型在运动过程中的变化矢量.采用Newton-Euler方程建立动力学方程式，将人机之间的交互力简化为弹力，根据运动中人体和外骨骼质心之间的距离变化得到其相对位移，从而求得运动过程中交互力的大小.最终在ADAMS(automatic dynamic analysis of mechanical system)仿真软件中对动力学模型进行仿真，并将动力学方程得到的关节力矩代入到仿真中，验证了该人体-外骨骼模型的正确性.     

Motion planning for planar bipeds using flatness: the single support phase

Trajectory planning and control of planar motions of biped robots is considered. The robot is modeled as a hierarchical structure of rigid links with rotational joints, which may be seen as a pendulum tree. Motors are available at all rotational joints. However, by the absence of control torques at the contact points with the ground, the system is underactuated. It is shown how differential flatness and time scaling can be helpful for the design of walking motions. Emphasis is put on the single support phase, when the robot touches the floor at a single point. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A damped bipedal inverted pendulum for human–structure interaction analysis

The bipedal inverted pendulum with damping has been adopted to simulate human–structure interaction recently. However, the lack of analysis and verification has provided motivation for further investigation. Leg damping and energy compensation strategy are required for the bipedal inverted pendulum to regulate gait patterns on vibrating structures. In this paper, the Hunt–Crossley model is adopted to get zeros contact force at touch down, while energy compensation is achieved by adjusting the stiffness and rest length of the legs. The damped bipedal inverted pendulum can achieve stable periodic gait with a lower energy input and flatter attack angle so that more gaits are available, compared to the template, referred to as spring-load inverted pendulum. The measured and simulated vertical ground reaction force-time histories are in good agreement. In addition, the dynamic load factors are also within a reasonable range. Parametric analysis shows that the damped bipedal inverted pendulum can achieve stable gaits of 1.6 to 2.4 Hz with a reasonable first harmonic dynamic load factor, which covers the normal walking step frequency. The proposed model in this paper can be applied to human–structure interaction analysis.     

Sensitivity Analysis of Human Leg Metabolical Costs

The human walking is characterized by skeletal dynamics and muscle excitation patterns minimizing the metabolical energy. This criterion is applied to assess the performance of lower limb prosthetic devices, and to evaluate therapies for patients presenting gait disorders. It is desirable, therefore, to dispose models of the human normal and pathological gaits capable of estimating the metabolical energy expenditure. For the swing phase of normal and pathological gaits a musculoskeletal model of the lower limb is presented to estimate metabolical energy expenditure. The mechanical model has three degrees of freedom and is actuated by eight Hill-type muscle units, and the model for the metabolical costs is adopted from literature. In this paper a combination of inverse and direct dynamics is used, and a sensitivity analysis of the dynamical behavior and the corresponding metabolical costs estimations with respect to parametrized neural excitations is performed. The leg motions are based on experiments in a gait analysis laboratory. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot

This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos.     

Modeling and identification of emotional aspects of locomotion

The studies of emotional facial expressions and emotional body language are currently receiving a lot of attention in the cognitive sciences. In this project, we study implicit bodily expression of emotions during standard motions, such as walking forwards.An underlying assumption of our work is that all human motion is optimal in some sense and that different emotions induce different objective functions, which result in different deformations of normal motion.We created a 3D rigid-body model of a human of which we use the forward dynamics simulation in an optimal control context. We performed two kinds of optimizations: (i) reconstruction of dynamic quantities, such as joint torques, of pre-recorded data of emotional walking motions and (ii) forward optimization that generates neutral and varied walking motions using different objective functions. Optimizations are performed with the software package MUSCOD-II, which uses a direct multiple-shooting discretization scheme. The results of this work form the foundation for further analysis of emotional motions using inverse optimal control methods.     

Mathematical simulation of a seven link biped robot on various surfaces and ZMP considerations

In recent years, numerous researches have been done based on simulation of legged mechanism, especially on biped robots simulation and control. The following article focuses on the biped robot simulation and control over various manners such as horizontal, ascending and descending surfaces with the aid of mathematical modeling methods (in MATLAB/SIMULINK environment). Similar parameters to human walking process will be obtained such as ZMP and joint’s actuator torques. The mathematical simulation has been used to interpolate trajectory of the robot path with the given break points. Of course, after the robot’s path determination, third-order spline method will be used because of the very high precision and ability to calculate the kinematic and dynamic parameters. With the aid of this program, common parameters such as linear and angular velocity and acceleration, joint’s angles and inertial forces for the given specifications and conditions (Nominal, no disturbances) will be calculated and simulated. Also, the two types of ZMP (Fixed and moving) have been considered and calculated with the aid of the software.     

Control of flexible joint robots using neural networks

The objective of this paper is to present a new controller designfor robot manipulators with elastic joints. The model, whichis used to represent the dynamics of elastic joint manipulators,is derived under two assumptions regarding the dynamic couplingbetween the actuators and the links. This model is useful forcases where the elasticity in the joints represents the dominantdynamic characteristic, and especially when it is of greatersignificance than gyroscopic interactions between the motorsand the links. The analysis of the problem is based on resultsin nonlinear control theory and particularly on the feedbacklinearization technique, and the controller design is achievedusing dynamic neural networks.     

Simulating Prosthetic Gait ‐ Lessons to Learn

Modern computer assisted above knee prostheses currently available on the market (e.g. C‐Leg from Otto‐Bock) allow the patients to choose their walking speed freely during level walking. Due to energy reasons it is only possible to change the passive mechanical characteristics of the prosthetic limb but no active actuators can be used to influence the knee joint movement. The control strategies for level walking act on the damping of the knee joint during swing phase of gait. During walking down ramps or descending stairs the control strategies are switched off in current designs. Based on a measured reference kinematics it should be possible to study the influence and behaving of technical devices in combination with the whole body biomechanical system. For this purpose we use a combined inverse‐forward‐dynamics approach. Kinematics parameters that are not critical in the inverse‐dynamics approach (leading to physiological reasonable forces and moments) and that are not influenced by the other parameters in the model are prescribed as in the classical inverse‐dynamics approach. To follow the other kinematics parameters (joint‐angles) control elements are used where the associated joint‐moments are determined by the goal trajectory and physiological boundary conditions. The so called “controlled joints” of the model and the applied technical device (prosthesis) are formulated as a forward‐dynamics system. In contrast to experimental studies a simulation study offers the advantage, that parameters can be varied in a wide range. Different ramp and stair inclinations can be implemented and different control strategies can be tested without any risk for the patient. The disadvantage lies in the fact, that there are only few reference sets for the kinematics of real patients. The modification of a reference kinematics for different situations (level walking, ramp or stair descent) is a big challenge and needs further investigation. On the other hand the combined inverse‐forward‐dynamics approach with its well balanced simplicity and complexity offers a flexible tool to reach this aim. This approach can also be used to investigate the interaction of the biomechanical system with other technical systems like sports equipment.     

Mathematical simulation of combined trajectory paths of a seven link biped robot

The following article focuses on biped robot simulation and control over combined trajectory paths with the aid of mathematical modeling methods focusing on the effects of hip height over torso’s modified motion. The mathematical simulation has been exploited to interpolate the combined trajectory of the robot path with the given breakpoints using inverse kinematic and dynamic methods to determine ZMP and stability treatments. After the robot’s combined path determination, a third-order spline is utilized because of its high precision and ability to calculate the kinematic, dynamic and control parameters. With the aid of this software, common parameters such as joint angles and inertial forces for the given specifications and nominal conditions are calculated and simulated.     

Nonlinear modeling and control of slosh in liquid container transfer via a PPR robot

This paper studies the point-to-point liquid container transfer control problem for a PPR robot. The robot manipulator is represented as three rigid links, and the liquid slosh dynamics are included using a multi-mass-spring model. It is assumed that two forces and a torque applied to the prismatic joints and the revolute joint, respectively, are available as control inputs. The objective is to control the robot end-effector movement while suppressing the sloshing modes. A nonlinear mathematical model that reflects all of these assumptions is first introduced. Then, Lyapunov-based feedback controllers are designed to achieve the control objective. Two cases are considered: partial-state feedback that does not use slosh state information and full-state feedback that uses both robot state and slosh state measurements or estimations. Computer simulations are included to illustrate the effectiveness of the proposed control laws.     

Effect of the deformability of structural links on the motion of a two-legged walking robot

The problem of the dynamics of parametric optimization of the motion of a walking robot is solved within the framework of a seven-link model with the elasticity of the structural components taken into account. The effect of the elasticity of the links on the kinematic, dynamic, and energy characteristics of the locomotion of the walking mechanism is studied.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 76–79, 1987.     

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.     

A simple model for clock-actuated legged locomotion

The spring-loaded inverted pendulum (SLIP) model describes well the steady-state center-of-mass motions of a diverse range of walking and running animals and robots. Here we ask whether the SLIP model can also explain the dynamic stability of these gaits, and we find that it cannot do so in many physically-relevant parameter ranges. We develop an actuated, lossy, clock-torqued SLIP, or CT-SLIP, with more realistic hip-motor torque inputs, that can capture the robust stability properties observed in most animals and some legged robots. Variations of CT-SLIP at a similar level of detail and complexity may also be appropriate for capturing the whole-system center-of-mass dynamics of locomotion of legged animals and robots varying widely in size and morphology. This paper contributes to a broader program to develop mathematical models, at varied levels of detail, that capture the dynamics of integrated organismal systems exhibiting integrated whole-body motion.      

Multifractal fluctuations in joint angles during infant spontaneous kicking reveal multiplicativity-driven coordination

Previous research has considered infant spontaneous kicking as a form of exploration. According to this view, spontaneous kicking provides information about motor degrees of freedom and may shape multijoint coordinations for more complex movement patterns such as gait. Recent work has demonstrated that multifractal, multiplicative fluctuations in exploratory movements index energy flows underlying perceptual-motor information. If infant spontaneous kicking is exploratory and occasions an upstream flow of information from the motor periphery, we expected not only that multiplicativity of fluctuations at the hip should promote multiplicativity of fluctuations at more distal joints (i.e., reflecting downstream effects of neural control) but also that multiplicativity at more distal joints should promote multiplicativity at the hip. Multifractal analysis demonstrated that infant spontaneous kicking in four typically developing infants for evidence of multiplicative fluctuations in multiple joint angles along the leg (i.e., hip, knee, and ankle) exhibited multiplicativity. Vector autoregressive modeling demonstrated that only one leg exhibited downstream effects but that both legs exhibited upstream effects. These results confirm the exploratory aspect of infant spontaneous kicking and suggest chaotic dynamics in motor coordination. They also resonate with existing models of chaos-controlled robotics and noise-based interventions for rehabilitating motor coordination in atypically developing patients.