Dynamic modeling,stability and energy consumption analysis of a realistic six-legged walking robot

This paper deals with a detailed analysis on kinematics, dynamics, stability and energy consumption of a realistic six-legged robot. The aim of this study is to extend a previous work of Roy et al. [1], in order to estimate optimal feet forces and joint torques of the six-legged robot generating wave-gaits with four different duty factors and deal with its stability issues. Two different approaches are developed to determine optimal feet forces. In the first approach, minimization of the norm of feet forces is carried out using a least square method, whereas minimization of the norm of joint torques is performed in the second approach. The second approach is found to be more energy efficient compared to the first one. The maximum values of feet forces and joint torques are seen to decrease with the increase of duty factor. The effects of walking parameters, namely velocity, stroke and duty factors have been studied on energy consumption and stability of the robot. The variations of average power consumption and specific energy consumption with the velocity and stroke are compared for four different duty factors. Wave gait with a low duty factor is found to be more energy-efficient compared to that with a high duty factor at the highest possible velocity.     

Effects of turning gait parameters on energy consumption and stability of a six-legged walking robot

Minimization of energy consumption plays a key role in the locomotion of a multi-legged robot used for various purposes. Turning gaits are the most general and important factors for omni-directional walking of a six-legged robot. This paper presents an analysis on energy consumption of a six-legged robot during its turning motion over a flat terrain. An energy consumption model is developed for statically stable wave gaits in order to minimize dissipating energy for optimal feet forces distributions. The effects of gait parameters, namely angular velocity, angular stroke and duty factors are studied on energy consumption, as the six-legged robot walks along a circular path of constant radius with wave gait. The variations of average power consumption and energy consumption per unit weight per unit traveled length with the angular velocity and angular stroke are compared for the turning gaits of a robot with four different duty factors. Computer simulations show that wave gait with a low duty factor is more energy-efficient compared to that with a high duty factor at the highest possible angular velocity. A stability analysis based on normalized energy stability margin is performed for turning motion of the robot with four duty factors for different angular strokes.     

Estimation of optimal feet forces and joint torques for on-line control of six-legged robot

The present study aims to estimate optimal feet forces and joint torques necessary for real-time control of a six-legged robot. Two approaches have been developed, such as minimization of norm of feet forces and minimization of norm of joint torques using the least squared method. Results of these two approaches have been compared with each other, and with those of available literature. As both of these approaches are found to be computationally fast, these are suitable for real-time control of the six-legged robot.     

Torque Distribution in a Six-Legged Robot

In this paper, distribution of required forces and moments to the supporting legs of a six-legged robot is handled as a torque-distribution problem. This approach is comparatively contrasted to the conventional approach of tip-point force distribution. The formulation of dynamics is performed by using the joint torques as the primary variables. The sum of the squares of the joint torques on the supporting legs is considered to be proportional to the dissipated power. The objective function is constructed as this sum, and the problem is formulated as to minimize this quadratic objective function with respect to linear equality and inequality constraints. It is demonstrated that the torque-distribution scheme results in a much more efficient distribution compared with the conventional scheme of force distribution. In contrast to the force distribution, the torque-distribution scheme makes good use of interaction forces and friction in order to minimize the required joint torques     

Dynamic balancing of planar mechanisms using toric geometry

A mechanism is statically balanced if for any motion, it does not apply forces on the base. Moreover, if it does not apply torques on the base, the mechanism is said to be dynamically balanced. In this paper, a new method for determining the complete set of dynamically balanced planar four-bar mechanisms is presented. Using complex variables to model the kinematics of the mechanism, the static and dynamic balancing constraints are written as algebraic equations over complex variables and joint angular velocities. After elimination of the joint angular velocity variables, the problem is formulated as a problem of factorization of Laurent polynomials. Using tools from toric geometry including toric polynomial division, necessary and sufficient conditions for static and dynamic balancing of planar four-bar mechanisms are derived.     

Soft computing-based expert systems to predict energy consumption and stability margin in turning gaits of six-legged robots

Turning gaits are the most general and very important ones for omni-directional walking of a six-legged robot. Soft computing-based expert systems have been developed in the present work to predict specific energy consumption and stability margin of turning gait of a six-legged robot. Besides back-propagation neural network, three approaches based on adaptive neuro-fuzzy inference system have been developed and their performances are compared with each other. Genetic algorithm-tuned multiple adaptive neuro-fuzzy inference systems are found to perform better than other approaches. This could be due to a more exhaustive search conducted by the genetic algorithm in place of back-propagation algorithm and the use of two separate adaptive neuro-fuzzy inference systems for two different outputs.     

Energy-optimal trajectory planning for car-like robots

When a battery-powered robot needs to operate for a long period of time, optimizing its energy consumption becomes critical. Driving motors are a major source of power consumption for mobile robots. In this paper, we study the problem of finding optimal paths and velocity profiles for car-like robots so as to minimize the energy consumed during motion. We start with an established model for energy consumption of DC motors. We first study the problem of finding the energy optimal velocity profiles, given a path for the robot. We present closed form solutions for the unconstrained case and for the case where there is a bound on maximum velocity. We then study a general problem of finding an energy optimal path along with a velocity profile, given a starting and goal position and orientation for the robot. Along the path, the instantaneous velocity of the robot may be bounded as a function of its turning radius, which in turn affects the energy consumption. Unlike minimum length paths, minimum energy paths may contain circular segments of varying radii. We show how to efficiently construct a graph which generalizes Dubins’ paths by including segments with arbitrary radii. Our algorithm uses the closed-form solution for the optimal velocity profiles as a subroutine to find the minimum energy trajectories, up to a fine discretization. We investigate the structure of energy-optimal paths and highlight instances where these paths deviate from the minimum length Dubins’ curves. In addition, we present a calibration method to find energy model parameters. Finally, we present results from experiments conducted on a custom-built robot for following optimal velocity profiles.     

Power Consumption Optimization for a Hexapod Walking Robot

Power consumption is one of the main operational restrictions on autonomous walking robots. In this paper, an energy efficiency analysis is performed for a hexapod walking robot to reduce these energy costs. To meet the power-saving demands of legged robots, the torque distribution algorithm required to minimize the system’s energy costs was established with an energy-consumption model formulated. In contrast to the force distribution method, where the objective function is related to the tip-point force components, the torque distribution scheme is based on minimization of the mechanical energy cost and heat loss power. The simulation results show that this scheme could reduce the system energy costs with use of the appropriate walking velocities and duty factors for the robot. The paper also discusses the effects of the gait patterns and the mechanical structure on the system energy costs. For this purpose, the prescribed periodic walking gait of the robot is described in terms of several parameters, including the duty factor, the stride length, the body height, and the foot trajectory lateral offset. The numerical results indicate some analogies between the characteristics of the simulated walking robot and those of animals in nature. The optimized parameters derived here are intended for robot platform development applications.     

Improving feasibility of robotic milling through robot placement optimisation

Milling performed with robots is quite demanding, even for low-strength materials, due to the high accuracy requirements, the generally high and periodically varying milling forces and the low stiffness of robots compared to CNC machine tools. In view of the generally improved recently robot stiffness, it is desirable to perform the milling operation in regions of the robot’s workspace where manipulability, both kinematic and dynamic, is highest, thereby exhausting the robot’s potential to cope with the process. In addition, by selecting the most suitable initial pose of the robot with respect to the workpiece, a reduction in the range of necessary joint torques may be reached, to the extent of alleviating the heavy requirements on the robot. Two genetic algorithms (GAs) are employed to tackle these problems. The values of several robot variables, such as joint positions and torques, which are needed by the genetic algorithms, are calculated using inverse kinematics and inverse dynamics models. In addition, initial positions and poses leading to singularities along the milling path are penalized and, thus, avoided. The first GA deals solely with robot kinematics to maximize manipulability. The second GA takes into account milling forces, which are computed numerically according to the particular milling parameters, to minimise joint torque loads.     

Dynamically balanced optimal gaits of a ditch-crossing biped robot

This paper deals with the generation of dynamically balanced gaits of a ditch-crossing biped robot having seven degrees of freedom (DOFs). Three different approaches, namely analytical, neural network (NN)-based and fuzzy logic (FL)-based, have been developed to solve the said problem. The former deals with the analytical modeling of the ditch-crossing gait of a biped robot, whereas the latter two approaches aim to maximize the dynamic balance margin of the robot and minimize the power consumption during locomotion, after satisfying a constraint stating that the changes of joint torques should lie within a pre-specified value to ensure its smooth walking. It is to be noted that the power consumption and dynamic balance of the robot are also dependent on the position of the masses on various links and the trajectory followed by the hip joint. A genetic algorithm (GA) is used to provide training off-line, to the NN-based and FL-based gait planners developed. Once optimized, the planners will be able to generate the optimal gaits on-line. Both the NN-based and FL-based gait planners are able to generate more balanced gaits and that, too, at the cost of lower power consumption compared to those yielded by the analytical approach. The NN-based and FL-based approaches are found to be more adaptive compared to the other approach in generating the gaits of the biped robot.     

Energy-optimal gait analysis of quadruped robots

It is important for walking robots such as quadruped robots to have an efficient gait. Since animals and insects are the basic models for most walking robots, their walking patterns are good examples. In this study, the walking energy consumption of a quadruped robot is analyzed and compared with natural animal gaits. Genetic algorithms have been applied to obtain the energy-optimal gait when the quadruped robot is walking with a set velocity. In this method, an individual in a population represents the walking pattern of the quadruped robot. The gait (individual) which consumes the least energy is considered to be the best gait (individual) in this study. The energy-optimal gait is analyzed at several walking velocities, since the amount of walking energy consumption changes if the walking velocity of the robot is changed. The results of this study can be used to decide what type of gait should be generated for a quadruped robot as its walking velocity changes. This work was presented, in part, at the Sixth International Symposium on Artificial Life and Robotics, Tokyo, Japan, January 15–17, 2001.     

Adaptive dynamic locomotion of quadrupedal robots with semicircular feet on uneven terrain

This paper proposes an adaptive trajectory generation method for quadruped robots with semicircular feet to control body speed and heading, and to minimize power consumption on uneven terrain. The semicircular foot with single line of contact is easier to solve the kinematic problems than flat feet with active ankle joints. And its wheel-like rolling motion leads to improvement in power autonomy and reduction of impact forces. The adaptive gait patterns are changed by the sequential modulation of the locomotion period and the stride per step which are determined by the desired body speed and heading commands, and external environments. The efficiency and performance of the developed method are verified through computer simulations and experiments using a hydraulic actuated quadrupedal robot.     

自由飘浮空间机器人系统基座姿态调整

规划机械臂的运动以调整作为其基座的卫星的姿态，既节约姿控燃料，又可作为常规姿控系统的备份手段．首先，建立自由飘浮空间机器人系统的状态方程，其状态变量为关节角和卫星姿态角，输入变量为关节角速度．基于系统能控性理论，规划连接系统初始状态和期望状态的路径，实现了仅通过机械臂关节的运动同时控制基座姿态和机械臂关节角的目的．从理论上分析了机械臂的能量消耗，给出了使能量指标最小的近似最优算法．仿真结果表明了该方法的有效性．     

Near-optimal gait generations of a two-legged robot on rough terrains using soft computing

A two-legged robot will have to generate its near-optimal gaits after ensuring maximum dynamic balance margin and minimum power consumption, while moving on the rough terrains containing some staircases and sloping surfaces. Moreover, the changes of joint torques should lie below a pre-specified small value to ensure its smooth walking. The balance of the robot and its power consumption are also dependent on hip trajectory and position of the masses on various limbs. Both neural network- and fuzzy logic-based gait planners have been developed for the same, the training of which are provided using a genetic algorithm off-line. Once optimized, the planners are found to generate optimal gaits of the two-legged robot successfully for the test cases.     

Design and Control of 6-DOF Mechanism for Twin-Frame Mobile Robot

A new lightweight six-legged robot that uses a simple mechanism and can move and work with high efficiency has been developed. This robot consists of two leg-bases with three legs each, and walks by moving each leg-base alternately. These leg-bases are connected to each other with a 6 degrees of freedom (DOF) mechanism. While designing this robot, the output force, velocity, and workspace of various connection mechanisms were compared, and the results showed that good performance could be achieved with a serial/parallel hybrid mechanism. The serial/parallel hybrid mechanism consists of three 6-DOF serially linked arms positioned with radial symmetry about the center of each leg-base; each leg-base is composed of two active and four passive joints. Walking experiments with this robot confirmed that this mechanism has satisfactory performance not only as a walking robot, but also as an active walking platform. Furthermore, in this robot, the entire leg-drive mechanism acts as a 6-axis force sensor, and individual sensors at the feet are not necessary. The forces and moments can be calculated from the changes in the joint angles. Experiments conducted verified that smooth contact with the ground by the swing-leg and successful switching from swing to support leg can be achieved using this force control and force measurement method.     

铰接式履带车辆转向特性仿真研究

针对深海富钴结壳行走装置必须适应复杂恶劣地形和较强的越障性能,提出一种具有四个液压缸的主动式联结装置的铰接式履带车辆,通过对铰接式履带车辆进行转向运动学分析,以及对联接装置的转向工作机理进行分析,得出了转向液压缸活塞杆伸长量与车辆理论转向半径的关系,并采用多体动力学仿真软件ADAMS的履带车辆工具箱ATV 建立了铰接式履带车辆的虚拟样机模型,仿真得到了铰接转向角速度,铰接转向半径,铰接转向所耗功率与差速转向所耗功率的关系,仿真结果表明,仿真模型具有较高的精度,能够对转向过程中主要参数随时间的变化历程进行仿真,可以为铰接式履带车辆转向过程的研究提供理论分析根据.     

Reducing Base Reactions With Gyroscopic Actuation of Space-Robotic Systems

In this paper, control-moment gyroscopes (CMGs) are proposed as actuators for a spacecraft-mounted robotic arm to reduce reaction forces and torques on the spacecraft base. With the established kinematics and dynamics for a CMG robotic system, numerical simulations are performed for a general CMG system with an added payload. The analysis of an added payload’s effects on otherwise reactionless CMG systems motivates the examination of possible operations concepts to reduce base reactions and power consumption. Simulation results for an example closed-loop maneuver show that base reactions can be significantly reduced, or even eliminated, with CMG actuation while using the same amount of power as a robotic system driven by conventional joint motors.      

Relation between global velocity and local torque optimization of redundant manipulators

In this article, the relation between the global optimization of joint velocity and local optimization of joint torque is investigated. The local minimization of the weighted joint torques can be matched to the global optimization of the corresponding weighted joint velocities when the weighted matrices satisfy certain sufficient conditions. A straightforward matching is obtained using the local optimization of the inertia inverse weighted dynamic torque and the global minimization of the kinetic energy. Another easy solution can be found, as will be shown later, if the inertia matrix is a constant and gravity is neglected. Based on that, it can be seen why a Cartesian robot, which has a constant inertia matrix, is always stable. © 1994 John Wiley & Sons, Inc.     

Torque Optimizing Control with Singularity-Robustness for Kinematically Redundant Robots

A new control method for kinematically redundant manipulators having the properties of torque-optimality and singularity-robustness is developed. A dynamic control equation, an equation of joint torques that should be satisfied to get the desired dynamic behavior of the end-effector, is formulated using the feedback linearization theory. The optimal control law is determined by locally optimizing an appropriate norm of joint torques using the weighted generalized inverses of the manipulator Jacobian-inertia product. In addition, the optimal control law is augmented with fictitious joint damping forces to stabilize the uncontrolled dynamics acting in the null-space of the Jacobian-inertia product. This paper also presents a new method for the robust handling of robot kinematic singularities in the context of joint torque optimization. Control of the end-effector motions in the neighborhood of a singular configuration is based on the use of the damped least-squares inverse of the Jacobian-inertia product. A damping factor as a function of the generalized dynamic manipulability measure is introduced to reduce the end-effector acceleration error caused by the damping. The proposed control method is applied to the numerical model of SNU-ERC 3-DOF planar direct-drive manipulator.     

Redundant robot control using task based performance measures

The joint velocities required to move the end-effector of a redundant robot with a desired linear and angular velocity depend on its configuration. Similarly, the joint torques produced due to the force and moment at the end-effector also depend on its configuration. When the robot is near a singular configuration, the joint velocities required to attain the end-effector velocity in certain directions are extremely high. Similarly, in some configurations the joint torque produced at certain joints may be high for a relatively small magnitude of external force. An infinite number of trajectories in the joint space can be used to achieve a desired end-effector trajectory for redundant robots. However, a joint trajectory resulting in robot configurations requiring lower joint velocities or joint torques is desired. This may be achieved through a proper utilization of redundancy. Local performance measures for redundant robots are defined in this article as indicators of their ability to follow a desired end-effector trajectory and their ability to apply desired forces at the end-effector. Thus, these performance measures depend on the task to be performed. Control algorithms which can be efficiently applied to redundant robots to improve these performance measures are presented. These control algorithms are based on the gradient projection method. Gradients of the performance measures used in the control schemes result in simple symbolic expressions for “real world” robots'. Feasibility and effectiveness of these control schemes is demonstrated through the simulation of a seven-degree-of-freedom redundant robot derived from the PUMA geometry.