Extension of the divide-and-conquer algorithm for the efficient inverse dynamics analysis of multibody systems

This paper presents a new mathematical framework to extend the Generalized Divide-and-Conquer Algorithm (GDCA) for the inverse dynamics analysis of fully actuated constrained multibody systems. Inverse-GDCA (iGDCA) is a highly parallelizable method which does not create the mass and Jacobian matrices of the entire system. In this technique, generalized driving forces and constraint loads due to kinematic pairs are clearly and separately differentiated from each other in the equations of motion. As such, it can be easily used for control scheme purposes. iGDCA works based on a series of recursive assembly and disassembly passes to form and solve the equations governing the inverse dynamics of the system. Herein, the mathematical formulations to efficiently combine the dynamics of consecutive bodies in the assembly pass for the purpose of inverse dynamics analysis are presented. This is followed by generating the disassembly pass algorithm to efficiently compute generalized actuating forces. Furthermore, this paper presents necessary mathematical formulations to efficiently treat the inverse dynamics of multibody systems involving kinematic loops with various active and passive boundary conditions. This is followed by the design of a new strategy to efficiently perform the assembly–disassembly pass in these complex systems while avoiding unnecessary computations. Finally, the presented method is applied to selected open-chain and closed-chain multibody systems.     

Optimal Path Programming of the Stewart Platform Manipulator Using the Boltzmann–Hamel–d'Alembert Dynamics Formulation Model

In this paper, the dynamics formulation of a general Stewart platform manipulator (SPM) with arbitrary geometry and inertia distribution is addressed. Based on a structured Boltzmann–Hamel–d'Alembert approach, in which the true coordinates are for translations and quasi-coordinates are for rotations, a systematic methodology using the parallelism inherent in the parallel mechanisms is developed to derive the explicit closed-form dynamic equations which are feasible for both forward and inverse dynamics analyses in the task space. Thus, a singularity-free path programming of the SPM for the minimum actuating forces is presented to demonstrate the applications of the developed dynamics model. Using a parametric path representation, the singularity-free path programming problem can be cast to the determination of undetermined control points, and then a particle swarm optimization algorithm is employed to determine the optimal control points and the associated trajectories. Numerical examples are implemented for the moving platform with constant orientations and varied orientations.     

Inverse dynamics of redundant manipulators using a minimum number of control forces

The number of the control actuators used by the inverse kinematics and dynamics algorithms that have been developed in the literature for generating redundant robot joint trajectories is equal to the number of the degrees of freedom of the manipulator. In this article, an inverse dynamics algorithm that performs the tasks using only a minimum number of actuators is proposed. The number of actuators is equal to the dimension of the task space, and the control forces are solved simultaneously with the corresponding system motion. It is shown that because all degrees of freedom are not actuated, the control forces may lose the ability to make an instantaneous effect on the end-effector acceleration at certain configurations, yielding the dynamical equation set of the system to be singular. The dynamical equations are modified in the neighborhood of the singular configurations by utilizing higher-order derivative information, so that the singularities in the numerical procedure are avoided. Asymptotically stable inverse dynamics closed-loop control in the presence of perturbations is also discussed. The algorithm is further generalized to closed chain manipulators. Three-link and two-link redundant planar manipulators are analyzed to illustrate the validity of the approach. © 2995 John Wiley & Sons, Inc.     

An algorithm for the inverse dynamics of n-axis general manipulators using Kane''s equations

Presented in this paper is an algorithm for the numerical solution of the inverse dynamics of robotic manipulators of the serial type, but otherwise arbitrary. The algorithm is applicable to manipulators containing n joints of the rotational or the prismatic type. For a given set of Hartenbeg-Denavit and inertial parameters, as well as for a given trajectory in the space of joint coordinates, the algorithm produces the time histories of the n torques or forces required to drive the manipulator through the prescribed trajectory. The algorithm is based on Kane's dynamical equations of mechanical multibody systems. Moreover, the complexity of the algorithm pressented here is lower than that of the most efficient inverse-dynamics algorithm reported in the literature. Finally, the applicability of the algorithm is illustrated with two fully solved examples.     

Dynamic formulation and performance evaluation of the redundant parallel manipulator

The dynamic formulation and performance evaluation of the redundant parallel manipulator are presented in this paper. By means of the principle of virtual work and the concept of link Jacobian matrices, the inverse dynamic model of the redundant parallel manipulator is set up. It consists of six linear consistent equations with eight unknown quantities. Then, the optimum solution of the actuating torques is achieved by employing the Moore-Penrose inverse matrix. It is with minimum norm and least quadratic sum among the possible actuating torque vectors. A series of new dynamic performance indices with obvious physical meanings have been proposed in the paper. By decoupling the inverse dynamics in the exhaustive way, a novel dynamic performance index combining the acceleration, velocity and gravity terms of the dynamic equations has been presented to evaluate the dynamic characteristic of the redundant parallel manipulator. With the index, it is possible to control the performance in the different direction. The index has been applied to the dynamic characteristic evaluation of the redundant parallel manipulator in the simulation. It is general and can be used for the dynamic performance evaluation of other types of parallel manipulators.     

An extended divide-and-conquer algorithm for a generalized class of multibody constraints

An extension to the divide-and-conquer algorithm (DCA) is presented in this paper to model constrained multibody systems. The constraints of interest are those applied to the system due to the inverse dynamics or control laws rather than the kinematically closed loops which have been studied in the literature. These imposed constraints are often expressed in terms of the generalized coordinates and speeds. A set of unknown generalized constraint forces must be considered in the equations of motion to enforce these algebraic constraints. In this paper dynamics of this class of multibody constrained systems is formulated using a Generalized-DCA. In this scheme, introducing dynamically equivalent forcing systems, each generalized constraint force is replaced by its dynamically equivalent spatial constraint force applied from the appropriate parent body to the associated child body at the connecting joint without violating the dynamics of the original system. The handle equations of motion are then formulated considering these dynamically equivalent spatial constraint forces. These equations in the GDCA scheme are used in the assembly and disassembly processes to solve for the states of the system, as well as the generalized constraint forces and/or Lagrange multipliers.     

A constrained motion perspective of railway vehicles collision

Impacts, friction, and normal contact forces occur in the railway vehicles couplers. This paper presents a novel nonsmooth model of the train collision; until now, only penalized models have been approached. The train dynamics is described by an equality of measures formulated at the velocity level. The equations of motion are integrated using the Moreau time-stepping algorithm. Impulsive and normal contact forces are described by a set-valued law of Signorini type, while friction forces are described by a set-valued law of Coulomb type. The constrained forces are computed deducing a particular, simplified formulation of the Udwadia–Kalaba equations. The resulting algorithm is simple and straightforward. Both impulsive and nonimpulsive dynamics are casted in the same framework. Any feature or situation regarding train collisions may be modeled. A demonstrative application is presented. Simulations reveal nonsmooth phenomena like simultaneous multiple collisions, stick-slip, captures, and offset in the final equilibrium position.     

Matrix modeling of inverse dynamics of spatial and planar parallel robots

Recursive matrix relations for kinematics and dynamics analysis of two known parallel mechanisms: the spatial 3-PRS and the planar 3-RRR are established in this paper. Knowing the motion of the platform, we develop first the inverse kinematical problem and determine the positions, velocities, and accelerations of the robot’s elements. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work, and the results can be verified in the framework of the Lagrange equations with their multipliers. Finally, compact matrix equations and graphs of simulation for power requirement comparison of each of three actuators in two different actuation schemes are obtained. For the same evolution of the moving platform, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems. The study of the dynamics of the parallel mechanisms is done mainly to solve successfully the control of the motion of such robotic systems.     

Experimental study of a momentum-based method for identifying the inertia barycentric parameters of a human body

Physical modeling, simulation and analysis of an individual human body require inertia properties of the body segments of the human. Such subject-specific inertia data can be obtained only by measuring the individual human body as opposed to be derived from statistically generated anthropometric database. This paper presents experimental validation of a momentum-based approach for identifying the barycentric parameters of an individual human body which fully describes the inertia properties of the human. The identification algorithm is derived from the impulse–momentum equations of the human body which is assumed to be a multibody system with tree-type topology. Since the impulse–momentum equations are linear in terms of the unknown barycentric parameters, these parameters can be solved from the equations using a least-squares method. The approach does not require measuring or estimating accelerations and joint forces/torques because they do not appear in the impulse–momentum equations, and thus, the resulting identification procedure is less demanding on measurement data than the methods derived from the equations of motion. In this paper the test results of the identification method are validated by comparing the identified inertia parameters against the statistically established anthropometric data. Additionally, the identification results are also confirmed by comparing the contact forces using inverse dynamics to those obtained by forces plates.     

冗余驱动并联机械手的混合位置/力自适应控制

针对冗余驱动并联机构研究一种自适应的混合位置/力控制算法.基于并联机构中约束子流形的几何性质,将冗余驱动并联机构的逆动力学自然投影到位形空间和约束力空间.基于投影方程,提出一种统一的具有渐进稳定性的自适应混合位置/力控制算法.采用最小二范数准则求解冗余解问题,实现了实际驱动关节力矩的优化.仿真结果验证了控制方法的有效性.     

Modeling and computational issues in the inverse dynamics simulation of triple jump

The triple jump is a demanding field event consisting of an approach run, and then followed by a hop, a bound, and a jump. The three consecutive takeoffs are executed at high speed, during which a jumper must absorb extremely large impact forces. The purpose of this paper is to develop an effective formulation for the inverse dynamics simulation of all the jump phases separately. A planar model of the jumper is used, composed of 14 rigid segments connected by 13 hinge joints, and actuated by muscle forces in the lower limbs and resultant muscle torques in the upper body joints. The equations of motion of the model are obtained using a projective technique, allowing for effective assessment of the ground reactions as well as muscle forces and joint reaction forces in the lower limbs. Some numerical results of the inverse dynamics simulation of a triple jump are reported.     

A Kane’s based algorithm for closed-form dynamic analysis of a new design of a 3RSS-S spherical parallel manipulator

This paper proposes a systematic methodology to obtain a closed-form formulation for dynamics analysis of a new design of a fully spherical robot that is called a 3(RSS)-S parallel manipulator with real co-axial actuated shafts. The proposed robot can completely rotate about a vertical axis and can be used in celestial orientation and rehabilitation applications. After describing the robot and its inverse position, velocity and acceleration analysis is performed. Next, based on Kane’s method, a methodology for deriving the dynamical equations of motion is developed. The elaborated approach shows that the inverse dynamics of the manipulator can be reduced to solving a system of three linear equations in three unknowns. Finally, a computational algorithm to solve the inverse dynamics of the manipulator is advised and several trajectories of the moving platform are simulated.

     

A unified dynamic algorithm for wheeled multibody systems with passive joints and nonholonomic constraints

This paper presents a systematic approach to develop a generalized symbolic/numerical dynamic algorithm for modeling and simulation of multibody systems with branches and wheels. The proposed dynamic algorithm includes the direct kinematic and inverse dynamic models of the wheeled systems with prismatic/revolute as well as actuated/passive degrees of freedom. Using the geometric configuration of the system through modified Denavit–Hartenberg convention, symbolic equations in general algorithmic form are developed for kinematic constraints associated with the wheel–ground contacts. The Newton–Euler equations are used to develop an algorithm for the inverse dynamic model of the multibody system. The complete algorithm is then used to solve the kinematics and dynamics of the system, and computes: (i) the kinematics of the external/internal passive degrees of freedom of the system, (ii) the Lagrange multipliers associated with the wheel–ground contacts, and (iii) the driving forces/torques of the actuated degrees of freedom. Some examples are solved with the help of the proposed algorithm, using MATLAB, to illustrate its implementation on different wheeled systems. These examples include a differential wheeled robot, a snake-like wheeled system, and a bicycle.     

Modeling and analysis of a multi-dimensional vibration isolator based on the parallel mechanism

A hybrid manipulator applied to vibration isolation of the manufacturing systems is proposed in this paper. The translations and rotations of the manipulator are decoupled, so the proposed isolator can isolate vibrations with wide range of frequency, at the same time it is fully capable of adjusting the orientations of the equipments. The scheme design, inverse kinematics, workspace and dexterity are carried out in this paper. A closed form dynamic model considering the external excitations on the base platform is performed based on the Newton–Euler approach. The optimum solutions of the forces in each actuating limb are obtained by using the Moor–Penrose inverse matrix. Furthermore, a novel dynamic performance index is proposed to evaluate the estimated maximum forces in the actuating limbs; this index can help to optimally design the parameters of motor, spring and damper. In order to evaluate the performance of isolation, the displacement transmissibility and acceleration transmissibility are also analyzed. The research work provides an analytical base for the development of the novel vibration isolator.     

Inverse dynamics of the 3-PRR planar parallel robot

Recursive modelling for the kinematics and dynamics of the known 3-PRR planar parallel robot is established in this paper. Three identical planar legs connecting to the moving platform are located in a vertical plane. Knowing the motion of the platform, we develop first the inverse kinematics and determine the positions, velocities and accelerations of the robot. Further, the principle of virtual work is used in the inverse dynamics problem. Several matrix equations offer iterative expressions and graphs for the power requirement comparison of each of three actuators in two different actuation schemes: prismatic actuators and revolute actuators. For the same evolution of the moving platform in the vertical plane, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems. The study of the dynamics of the parallel mechanisms is done mainly to solve successfully the control of the motion of such robotic systems.     

Hybrid force and motion control of flexible joint parallel manipulators using inverse dynamics approach

An inverse dynamics control algorithm is developed for hybrid motion and contact force trajectory tracking control of flexible joint parallel manipulators. First, an open-tree structure is considered by the disconnection of adequate number of unactuated joints. The loop closure constraint equations are then included. Elimination of the joint reaction forces and the other intermediate variables yield a fourth-order relation between the actuator torques and the end-effector position and contact force variables, showing that the control torques do not have an instantaneous effect on the end-effector contact forces and accelerations because of the flexibility. The proposed control law provides simultaneous and asymptotically stable control of the end-effector contact forces and the motion along the constraint surfaces by utilizing the feedback of positions and velocities of the actuated joints and rotors. A two degree of freedom planar parallel manipulator is considered as an example to illustrate the effectiveness of the method.     

基于动力学与控制统一模型的蛇形机器人速度跟踪控制方法研究

对带有被动轮的蛇形机器人进行速度跟踪控制时,利用传统的动力学建模方法得到的动力学方程复杂且不利于控制器的设计. 本文基于微分几何的方法将带有被动轮的蛇形机器人动力学投影到速度分布空间中, 得到了动力学与控制统一模型, 更有利于速度跟踪控制器的设计. 考虑到蛇形机器人在进行速度跟踪时容易出现奇异位形, 提出增加头部扰动速度的方法. 基于头部扰动速度和统一模型, 提出避免奇异位形的速度跟踪控制方法, 最后通过逆向动力学得到控制力矩. 文中对速度跟踪控制进行了数值仿真和实验验证. 仿真和实验结果表明, 提出的速度跟踪控制方法能够跟踪想要方向的速度, 并且在跟踪过程中可以有效地避免奇异位形.     

Contact Modeling and Identification of Planar Somersaults on the Trampoline

This paper presents an extensive study on the trampoline-performed planar somersaults. First, a multibody biomechanical model of the trampolinist and the recurrently interacting trampoline bed are developed, including both the motion equations and the determination of joint reactions. The mathematical model is then identified –the mass and inertia characteristics of the human body are estimated, and the stiffness and damping characteristics of the trampoline bed are measured. By recording the actual somersault performances the motion characteristics of the stunts, i.e. the time variations of positions, velocities and accelerations of the body parts are also obtained. Finally, an inverse dynamics formulation for the system designated as an under-controlled system, is developed. The followed inverse dynamics simulation results in the torques of muscle forces in the joints that assure the realization of the actual motion. The reaction forces in the joints during the analyzed evolutions are also determined. Using the kinematic and dynamics characteristics, the nature of the stunts, the way the human body is maneuvered and controlled, can be studied.     

Dynamics modelling of a Stewart-based hybrid parallel robot

Matrix relations for kinematics and dynamics analysis of a spatial two-module Stewart-based hybrid parallel robot are established in this paper. Knowing the relative motions of two moving platforms, the inverse dynamics problem is solved based on a set of recursive explicit equations. Finally, compact results and graphs of simulation for the input forces and powers of all actuators are obtained.