A rotationless formulation of multibody dynamics: Modeling of screw joints and incorporation of control constraints

In the present work, a new energy-momentum conserving time-stepping scheme for multibody systems comprising screw joints is developed. In particular, it is shown that the underlying rotationless formulation of multibody dynamics along with a specific coordinate augmentation technique makes possible the energy-momentum discretization of the screw pair. In addition to that, control (or servo) constraints are treated within the rotationless framework of multibody dynamics. The control constraints are used to partially prescribe the motion of a multibody system. In particular, control constraints, in conjunction with the coordinate augmentation technique, make possible to solve inverse dynamics problems by applying the present simulation approach.     

Biomechanics of manual material handling through simulation: Computational aspects

Computerized human motion simulation allows generation of dynamic human motions on computers. Biomechanical stresses can be estimated using the motions generated on a computer without actually collecting joint coordinate data. A two-dimensional whole-body lifting simulation model is presented in this paper. The model assumes that humans perform lifting activities based on minimization of physical work, subject to various constraints. The simulation method contains three major computation units: trajectory formation unit, dynamics of motion unit, and nonlinear optimization unit. The trajectory formation unit generates smooth polynomials representing motion characteristics of human lifting. Kinematics and kinetics are calculated in the dynamics unit. Objective and constraint functions are evaluated in the optimization unit. Optimal motions are generated by minimizing the objective function, subject to the constraints. Computation methods of the three units and simulation results are presented.     

A unified approach for inverse and direct dynamics of constrained multibody systems based on linear projection operator: applications to control and simulation

This paper presents a unified approach for inverse and direct dynamics of constrained multibody systems that can serve as a basis for analysis, simulation, and control. The main advantage of the dynamics formulation is that it does not require the constraint equations to be linearly independent. Thus, a simulation may proceed even in the presence of redundant constraints or singular configurations, and a controller does not need to change its structure whenever the mechanical system changes its topology or number of degrees of freedom. A motion-control scheme is proposed based on a projected inverse-dynamics scheme which proves to be stable and minimizes the weighted Euclidean norm of the actuation force. The projection-based control scheme is further developed for constrained systems, e.g., parallel manipulators, which have some joints with no actuators (passive joints). This is complemented by the development of constraint force control. A condition on the inertia matrix resulting in a decoupled mechanical system is analytically derived that simplifies the implementation of the force control. Finally, numerical and experimental results obtained from dynamic simulation and control of constrained mechanical systems, based on the proposed inverse and direct dynamics formulations, are documented.     

Optimum motion planning in joint space for robots using genetic algorithms

The optimum motion planning in joint space (OMPJS) for robots, which generally consists of two subproblems, optimum path planning and optimum trajectory planning, was considered as a whole in the paper. A new method for optimum motion planning problem based on an improved genetic algorithm is proposed, which is more general, flexible and effective. This approach incorporates kinematics constraints, dynamics constraints, and control constraints of robotic manipulator. The simulation results for a two and a three degrees of freedom robots are presented and discussed. The simulations are based on genetic algorithm class library WGAClass 1.0 developed by us with Borland C++ 3.1.     

Contouring control of a 3-[P2(US)] parallel manipulator

A three-translational degree of freedom parallel mechanism is introduced in this research. Kinematic and dynamic modelling of the parallel mechanism is investigated and equations of motion are derived via Lagrange formulation. Reduced dynamic equations of the robot are represented as a matrix form for implementation in inverse dynamics control of the constrained system. Equivalent contouring errors are represented in moving tangent-normal coordinates system in joint space. Contouring control of the robot is applied to the system using inverse-dynamics-based feedback linearization technique and application of the control technique is represented through two proposed manoeuvres for robot motion. Trajectory planning of the end-effector path is performed by specification of some accuracy points and cubic spline interpolation through them. Response performance in contour following of the control system is improved by appropriate choice of control parameters and by decoupling of error dynamics via assigning suitable structures for control matrices. The results of practical implementation of the control technique show proper accuracy of the control system in following two desired trajectories and acceptable errors are created with respect to the desired contours.     

Kinematic design of a redundant parallel mechanism for maskless lithography optical instrument manipulations

A new precision parallel mechanism having actuation redundancy will be introduced in this paper. Physical contribution of the actuation redundancy for the precision parallel mechanism is reviewed. In addition, several kinematic configurations have been analyzed for degrees of freedom verification and actuation redundancy. A new kinematic configuration which is 4-[P P]PS is suggested. The suggested 4-[P P]PS mechanism which has actuation redundancy provides six degrees of freedom to the mobile platform. For position control and path planning of the mobile platform, the inverse and the forward kinematics are solved for closed-form solutions. In order to verify the inverse and the forward kinematics, a numerical simulation result is presented. In addition to the inverse and forward accuracy proof, the numerical analysis provides other information such as independent translation motion, calibrated rotation arm at tilting motion, and symmetric motion at rotating motion.     

An efficient motion generation method for redundant humanoid robot arms based on motion continuity

This paper proposes a novel method of motion generation for redundant humanoid robot arms, which can efficiently generate continuous collision-free arm motion for the preplanned hand trajectory. The proposed method generates the whole arm motion first and then computes the actuators’ motion, which is different from IK (inverse kinematics)-based motion generation methods. Based on the geometric constraints of the preplanned trajectory and the geometric structure of humanoid robot arms, the wrist trajectory and elbow trajectory can be got first without solving inverse kinematics and forward kinematics. Meanwhile, the constraints restrict all feasible arm configurations to an elbow-circle and reduce the arm configuration space to a two-dimension space. By combining the configuration space and collision distribution of arm motion, collision-free arm configurations can be identified and be used to generate collision-free arm motion, which can avoid unnecessary forward and inverse kinematics. The experiments show that the proposed method can generate continuous and collision-free arm motion for preplanned hand trajectories.     

Index reduction by minimal extension for the inverse dynamics simulation of cranes

The present work deals with the inverse dynamics simulation of underactuated mechanical systems relying on servo constraints. The servo-constraint problem of discrete mechanical systems is governed by differential–algebraic equations (DAEs) with high index. We propose a new index reduction approach, which makes possible the stable numerical integration of the DAEs. The new method is developed in the framework of a specific crane formulation and facilitates a reduction from index five to index three and even to index one. Particular attention is placed on the special case in which the reduced index-1 formulation is purely algebraic. In this case the system at hand can be classified as differentially flat system. Both redundant coordinates and minimal coordinates can be employed within the newly developed approach. The success of the proposed method is demonstrated with two representative numerical examples.     

Human lifting simulation using a multi-objective optimization approach

This paper presents a multiobjective optimization (MOO) approach to predicting dynamic lifting for a three-dimensional, highly redundant digital human model with 55 degrees of freedom. The optimization problem is formulated to optimize two objective functions simultaneously—dynamic effort and stability—subject to basic physical and kinematical constraints. The predictive dynamics approach is used to solve for the joint angles, torque profiles, and ground reaction forces. The weighted sum approach of MOO is used to aggregate the two objective functions, and the Pareto optimal set for the problem is generated by systematically varying the weighting parameters for the objective functions. Experimental data are used to validate the final simulation. Several examples are presented to demonstrate the effect of the weighting parameters for the two objective functions on the predicted box-lifting strategies. The results show that the proposed MOO approach improves the simulation results compared to the single objective optimization formulation. Also, the formulation is less sensitive to the weighting coefficient for the stability criterion.     

Modular and Recursive Kinematics and Dynamics for Parallel Manipulators

Constrained multibody systems typically feature multiple closed kinematic loops that constrain the relative motions and forces within the system. Typically, such systems possess far more articulated degrees-of-freedom (within the chains) than overall end-effector degrees-of-freedom.Thus, actuation of a subset of the articulations creates mixture of active and passive joints within the chain.The presence of such passive joints interferes with the effective modular formulation of the dynamic equations-of-motion in terms of a minimal set of actuator coordinates as well the subsequent recursivesolution for both forward and inverse dynamics applications. Thus, in this paper, we examine the development of modular and recursive formulations of equations-of-motion in terms of a minimal set of actuated-joint-coordinates for an exactly-actuated parallel manipulators. The 3 RRR planar parallel manipulator, selected to serve as a case-study, is an illustrative example of a multi-loop, multi-degree-of-freedom system with mixtures of active/passive joints. The concept of decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism inherent in parallel mechanisms to develop a dynamics formulation that is both recursive and modular. An algorithmic approach to the development of both forward and inverse dynamics is highlighted. The presented simulation studies highlight the overall good numerical behavior of the developed formulation, both in terms of accuracy and lack of formulation stiffness.     

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.     

A Systematic Approach for Designing Analytical Dynamics and Servo Control of Constrained Mechanical Systems

A systematic approach for designing analytical dynamics and servo control of constrained mechanical systems is proposed. Fundamental equation of constrained mechanical systems is first obtained according to Udwadia-Kalaba approach which is applicable to holonomic and nonholonomic constrained systems no matter whether they satisfy the D'Alember's principle. The performance specifications are modeled as servo constraints. Constraint-following servo control is used to realize the servo constraints. For this inverse dynamics control problem, the determination of control inputs is based on the Moore-Penrose generalized inverse to complete the specified motion. Secondorder constraints are used in the dynamics and servo control. Constraint violation suppression methods can be adopted to eliminate constraint drift in the numerical simulation. Furthermore, this proposed approach is applicable to not only fully actuated but also underactuated and redundantly actuated mechanical systems. Two-mass spring system and coordinated robot system are presented as examples for illustration.      

Modeling and simulation of closed loop multibody systems with bodies-joints composite modules

Modular approaches are effective in improving the modeling efficiency in multibody system dynamics. A modular method, called bodies-joints composite simulation (BJCS) is presented in this paper. Two types of bodies-joints composite modules, i.e., f modules and 0 modules, are defined according to topology design rules of closed loop mechanisms. By this module partitioning, the differential-algebraic equations of motion of the system can be separated into purely algebraic and differential equations by structure decomposition. The stability criterion for the simulation is derived and a closed-form formulation to solve the algebraic loop problem is proposed. Simulation results of the forward dynamics of a 5R mechanism and a 6-UPS platform are presented to show the feasibility of the method. The CPU times of these two case studies are provided and compared with the generalized coordinate partitioning of Wehage and Haug and the modular simulation method of Kübler and Schiehlen, which indicate that the closed-form algebraic loop solver is more efficient than the numerical ones.     

仿人机器人复杂动作设计中人体运动数据提取及分析方法

提出了仿人机器人复杂动作设计中人体运动数据提取及分析方法. 首先, 通过运动捕捉系统获取人体运动数据, 并采用运动重定向技术, 输出人--机简化模型的数据; 然后, 对运动数据进行分析和运动学解算, 给出基于人体运动数据的仿人机器人逆运动学求解方法, 得到仿人机器人模型的关节角数据; 再经过运动学约束和稳定性调节后, 生成能够应用于仿人机器人的运动轨迹. 最终, 通过在仿人机器人BHR-2上进行刀术实验验证了该方法的有效性.     

Improved DAE formulation for inverse dynamics simulation of cranes

Cranes are underactuated systems with less control inputs than degrees of freedom. Dynamics and control of such systems is a challenging task, and the existence of solution to the inverse dynamics simulation problem in which an r-degree-of-freedom system with m actuators, m<r, is subject to m specified motion task (servo-constraints) is conditioned upon the system is differentially flat (all the system states and control inputs can be algebraically expressed in terms of the outputs and their time derivatives up to a certain order). The outputs are often designed as specified in time load coordinates to model a rest-to-rest maneuver along a trajectory in the working space, from the initial load position to its desired destination. The flatness-based methodology results then in the required control inputs determined in terms of the fourth time derivatives of the imposed outputs, and the derivations are featured by substantial complexity. The DAE formulation motivated in this contribution offers a more convenient approach to the prediction of dynamics and control of cranes executing prescribed load motions, and only the second time derivatives of the specified outputs are involved. While most of the inverse simulation formulations, both flatness-based and DAE ones, are performed using independent state variables, the use of dependent coordinates and velocities may lead to substantial modeling simplifications and gains in computational efficiency. An improved DAE formulation of this type is presented in this paper.     

The principles and practical application of helicopter inverse simulation

Inverse simulation is a technique whereby the control actions required for a modelled vehicle to fly a specified manoeuvre can be established. In this paper the general concepts of inverse simulation are introduced, and an algorithm designed specifically to achieve inverse simulation of a single main and tail rotor helicopter is presented. An important element of an inverse simulation is the design of the input functions i.e. manoeuvre definitions, and the methods used are also detailed. A helicopter mathematical model is also discussed along with the validation and verification of the inverse simulation. Finally, the applicability of the method is demonstrated by illustration of its use in two flight dynamics studies.     

Modeling Spatial Motion of 3D Deformable Multibody Systems with Nonlinearities

A computational strategy for modeling spatial motion of systems of flexible spatial bodies is presented. A new integral formulation of constraints is used in the context of the floating frame of reference approach. We discuss techniques to linearize the equations of motion both with respect to the kinematical coupling between the deformation and rigid body degrees of freedom and with respect to the geometrical nonlinearities (inclusion of stiffening terms). The plastic behavior of bodies is treated by means of plastic multipliers found as the result of fixed-point type iterations within a time step. The time integration is based on implicit Runge Kutta schemes with arbitrary order and of the RadauIIA type. The numerical results show efficiency of the developed techniques.     

Nonlinear inverse optimization approach for determining the weights of objective function in standing reach tasks

This paper presents a nonlinear inverse optimization approach to determine the weights for the joint displacement function in standing reach tasks. This inverse optimization problem can be formulated as a bi-level highly nonlinear optimization problem. The design variables are the weights of a cost function. The cost function is the weighted summation of the differences between two sets of joint angles (predicted posture and the actual standing reach posture). Constraints include the normalized weights within limits and an inner optimization problem to solve for joint angles (predicted standing reach posture). The weight linear equality constraints, obtained through observations, are also implemented in the formulation to test the method. A 52 degree-of-freedom (DOF) human whole body model is used to study the formulation and visualize the prediction. An in-house motion capture system is used to obtain the actual standing reach posture. A total of 12 subjects (three subjects for each percentile in stature of 5th percentile female, 50th percentile female, 50th percentile male and 95th percentile male) are selected to run the experiment for 30 tasks. Among these subjects one is Turkish, two are Chinese, and the rest subjects are Americans. Three sets of weights for the general standing reach tasks are obtained for the three zones by averaging all weights in each zone for all subjects and all tasks. Based on the obtained sets of weights, the predicted standing reach postures found using the direct optimization-based approach have good correlation with the experimental results. Sensitivity of the formulation has also been investigated in this study. The presented formulation can be used to determine the weights of cost function within any multi-objective optimization (MOO) problems such as any types of posture prediction and motion prediction.     

Nonlinear inverse optimization approach for determining the weights of objective function in standing reach tasks

This paper presents a nonlinear inverse optimization approach to determine the weights for the joint displacement function in standing reach tasks. This inverse optimization problem can be formulated as a bi-level highly nonlinear optimization problem. The design variables are the weights of a cost function. The cost function is the weighted summation of the differences between two sets of joint angles (predicted posture and the actual standing reach posture). Constraints include the normalized weights within limits and an inner optimization problem to solve for joint angles (predicted standing reach posture). The weight linear equality constraints, obtained through observations, are also implemented in the formulation to test the method. A 52 degree-of-freedom (DOF) human whole body model is used to study the formulation and visualize the prediction. An in-house motion capture system is used to obtain the actual standing reach posture. A total of 12 subjects (three subjects for each percentile in stature of 5th percentile female, 50th percentile female, 50th percentile male and 95th percentile male) are selected to run the experiment for 30 tasks. Among these subjects one is Turkish, two are Chinese, and the rest subjects are Americans. Three sets of weights for the general standing reach tasks are obtained for the three zones by averaging all weights in each zone for all subjects and all tasks. Based on the obtained sets of weights, the predicted standing reach postures found using the direct optimization-based approach have good correlation with the experimental results. Sensitivity of the formulation has also been investigated in this study. The presented formulation can be used to determine the weights of cost function within any multi-objective optimization (MOO) problems such as any types of posture prediction and motion prediction.