Closed-form Dynamic Equations Research Articles

Non-Slender n-Link Chain Driven by Single-Joint and Multi-Joint Muscle Actuators: Closed-Form Dynamic Equations and Joint Reaction Forces

The author has derived the closed-form dynamic equations for a planar musculoskeletal chain composed of a generic number n of rigid links connected by ideal revolute joints. Single-joint and multi-joint muscles have been modeled as linear force actuators that can span from one joint to all the joints of the chain. The generic shape and size of each individual link of the chain accounts for different alignments among the center of mass of the link, the centers of rotation of the joints that articulate the link with its neighbors, and the points of application of the muscle forces and the possible contact external resistances acting on the link. The joint torque and the reaction force acting on each joint have been determined in closed-form by analytical quantification of the unique contribution of each individual kinematic and kinetic variable: (1) force of each single-joint or multi-joint muscle spanning or non-spanning the joint; (2) weight and contact external resistances acting on each individual link of the chain; (3) position, angular velocity, and angular acceleration of each individual link of the chain. The analytical results derived in this study can be applied to multilink musculoskeletal chains with deep/superficial and segmental/global muscles.

Reduced Dynamic Modeling for Heavy-Duty Hydraulic Manipulators With Multi-Closed-Loop Mechanisms

A reduced dynamic modeling approach is introduced to systematically establish explicit closed-form dynamic equations for the main motion system of a heavy-duty hydraulic manipulator with multi-closed-loop mechanisms. The harmonious combination of the reduced system dynamic method with Lagrangian formulation, the principle of virtual work and screw theory greatly reduces the tedious calculation and largely simplifies the derivation of explicit control-orientated closed-form dynamic equations for complex multi-closed-loop mechanisms. Only three coupled subsystems, two Jacobian matrices, and two Hessian matrices are involved, thereby greatly reducing the order and the complexity of the closed-form dynamic equations. In addition to calculating the two Jacobian matrices by screw theory, the two Hessian matrices are also calculated straightforwardly by screw theory, thereby avoiding the difficulty in obtaining Hessian matrices by differentiating the Jacobian matrices and simplifying the calculation of the two Hessian matrices. No parts of dynamic equations are neglected in the derivation of the dynamic model. Thus, the accurate dynamic motion equations for the main motion system are obtained concisely. The derived closed-form dynamic equations are explicit with respect to the system inputs, which facilitate dynamics analysis and controller design. The experiments on the main motion system of the heavy-duty hydraulic forging manipulator demonstrate the efficiency of the proposed approach.

Closed-Form Dynamic Formulation of a General 6-P US Robot

The 6-P US parallel manipulator can be used as a good alternative for the 6-UP S manipulator, known as Stewart platform. 6-P US parallel robots may be designed and manufactured in various architectures with different attributes. In this paper, to assess all the possible architectures of the 6-P US with respect to both workspace and dynamic performance, a general geometry is first defined. Using Newton-Euler method, the dynamic model of the general 6-P US robot is derived and its closed-form dynamic equations are presented. More accurate formulation is obtained by considering all robot’s component inertia as well as including the angular velocity and acceleration vectors of the robot’s legs. Moreover, the effect of neglecting link inertia of the 6-P US robot is studied for different payload to link mass ratios. The closed-form model includes dynamic matrices of the robot which can conveniently be used for model based control techniques. Two trajectories are considered and the derived dynamic formulation is verified using a commercial multibody dynamics software. Finally, four case studies covering the well-known architectures of the 6-P US manipulator, including the Hexaglide and the HexaM, are compared based on the robot workspace and forces of the actuators. The results indicates that amongst the studied designs of the 6-P US robot, the architecture with rails leaning outside provides the best performance and therefore can be considered as a competitor for the conventional Stewart platforms.

Closed-form dynamics of a 3-DOF spatial parallel manipulator by combining the Lagrangian formulation with the virtual work principle

Dynamic modeling of spatial multi-degree-of-freedom (multi-DOF) parallel motion system is a challenging research because of its multi-closed-loop configuration, in contrast to serial manipulator. This study readdresses the issue of deriving explicit closed-form dynamic equations in the actuation space of non-redundant parallel mechanisms through the Lagrange modeling approach. Due to the complex relationship between passive joint variables and active joint variables, the application of the Lagrangian formulation for dynamic modeling of parallel mechanisms is considered nearly impossible. In this paper, the utilization of the virtual work principle makes the application of the Lagrangian formulation for parallel mechanisms possible and efficient. A closed-loop parallel mechanism is divided into several serial open-loop subchains. Explicit dynamic equations of each subchain can be derived by using the Lagrangian formalism straightforwardly with respect to their own local generalized coordinates. The principle of virtual work is used to combine the differential dynamic equations of the subsystems into an explicit dynamic model of the full parallel manipulator with respect to robot’s active generalized coordinates. The Jacobian and Hessian matrices play a crucial role in the transformation of the dynamic equations with respect to different generalized coordinates due to the application of the virtual work principle. The introduction of the cubic Hessian matrices makes the dynamic equations more compact. The procedure of modeling for a 3-DOF spatial parallel manipulator is taken as an application example of the proposed approach. The efficiency of the proposed approach and the correctness of the dynamic model are demonstrated by simulations and experiments.

Dynamic Modeling of a Six Degree-of-Freedom Flight Simulator Motion Base

This paper presents a closed-form solution for the direct dynamic model of a flight simulator motion base. The motion base consists of a six degree-of-freedom (6DOF) Stewart platform robotic manipulator driven by electromechanical actuators. The dynamic model is derived using the Newton–Euler method. Our derivation is closed to that of Dasgupta and Mruthyunjaya (1998, “Closed Form Dynamic Equations of the General Stewart Platform Through the Newton–Euler Approach,” Mech. Mach. Theory, 33(7), pp. 993–1012), however, we give some insights into the structure and properties of those equations, i.e., a kinematic model of the universal joint, inclusion of electromechanical actuator dynamics and the full dynamic equations in matrix form in terms of Euler angles and platform position vector. These expressions are interesting for control, simulation, and design of flight simulators motion bases. Development of a inverse dynamic control law by using coefficients matrices of dynamic equation and real aircraft trajectories are implemented and simulation results are also presented.

Comments to the: “Closed-form dynamic equations of the general Stewart platform through the Newton–Euler approach” and “A Newton–Euler formulation for the inverse dynamics of the Stewart platform manipulator”

Abstract Considering the novelty of the method and creativity of the research that is detailed in the two papers [Mechanism and Machine Theory, Vol. 33, No.7, pp. 993–1012, 1998 and Vol. 33, No.8, pp.1135–1152, 1998], we believe that correcting the following errors appearing in these two papers would enhance their valuable contribution to this field of research.

A Constrained Lagrange Formulation of Multilink Planar Flexible Manipulator

In this article, the closed-form dynamic equations of planar flexible link manipulators (FLMs), with revolute joints and constant cross sections, are derived combining Lagrange’s equations and the assumed mode shape method. To overcome the lengthy and complicated derivative calculation of the Lagrangian function of a FLM, these computations are done only once for a single flexible link manipulator with a moving base (SFLMB). Employing the Lagrange multipliers and the dynamic equations of the SFLMB, the equations of motion of the FLM are derived in terms of the dependent generalized coordinates. To obtain the closed-form dynamic equations of the FLM in terms of the independent generalized coordinates, the natural orthogonal complement of the Jacobian constraint matrix, which is associated with the velocity constraints in the linear homogeneous form, is used. To verify the proposed closed-form dynamic model, the simulation results obtained from the model were compared with the results of the full nonlinear finite element analysis. These comparisons showed sound agreement. One of the main advantages of this approach is that the derived dynamic model can be used for the model based end-effector control and the vibration suppression of planar FLMs.

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.

WITHDRAWN: Commentson ‘Closed form dynamic equations of the general Stewart platform through the Newton–Euler approach’ by B. Dasgupta and T.S. Mruthyunjaya [Mech. Mach. Theory 33(1998) 993–1012] and ‘A Newton–Eulerformulation for the inverse dynamics of the Stewart platform manipulator’ by B. Dasgupta and T.S. Mruthyunjaya [Mech. Mach. Theory 33(1998) 1135–1152

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A Model Reduction Method for the Control of Rigid Mechanisms

This paper presents a reduction method to build closed-form dynamic equations for rigid multibody systems with a minimal kinematic description. Relying on an initial parameterization with absolute displacements and rotations, the method is able to tackle complex topologies with closed-loops in a systematic way and its extension to flexible multibody systems will be investigated in the future. Thus, it would be of great use in the framework of model-based control of mechanisms. The method is based on an interpolation strategy. The initial model is built and reduced for a number of selected points in the configuration space. Then, a piecewise polynomial model is adjusted to match the collected data. After the presentation of the reduction procedure and of the interpolation strategy, two applications of the reduction method are considered: a four-bar mechanism and a parallel kinematic machine-tool called “Orthoglide”.

Dynamic analysis and simulation of a six degree of freedom Stewart platform manipulator

This article presents the explicit compact closed-form dynamic equations in the task-space by applying the combination of the Newton—Euler method with the Lagrange formulation including the dynamics of the legs for the Stewart platform manipulator. The kinematics analysis of the manipulator is given and the velocity and the acceleration formulae needed to derive the dynamic equations are also derived. The driving forces acting on the legs are determined according to the dynamic formulation. The formulation has been implemented in routines and has been used for studying a few inverse dynamic problems of a specific Stewart platform manipulator. Simulation results reveal the effect of the leg inertia and that of its parts, respectively, on the dynamics of the complete system, and numerical examples show the effectiveness of the proposed method and the dynamic equations of the Stewart platform manipulator.

Coordinate-invariant algorithms for robot dynamics

We present, using methods from the theory of Lie groups and Lie algebras, a coordinate-invariant formulation of the dynamics of open kinematic chains. We first re-formulate the recursive dynamics algorithm originally given by Park et al. (1995) for open chains in terms of standard linear operators on the Lie algebra of the special Euclidean group. Using straight forward algebraic manipulations, we then recast the resulting algorithm into a set of closed-form dynamic equations. We then reformulate Featherstone's (1987) articulated body inertia algorithm using this same geometric framework, and re-derive Rodriguez et al.'s (1991, 1992) square factorization of the mass matrix and its inverse. An efficient O(n) recursive algorithm for forward dynamics is also extracted from the inverse factorization. The resulting equations lead to a succinct high-level description of robot dynamics in both joint and operational space coordinates that minimizes symbolic complexity without sacrificing computational efficiency, and provides the basis for a dynamics formulation that does not require link reference frames in the description of the forward kinematics.

Closed-Form Dynamic Equations of the General Stewart Platform through the Newton–Euler Approach

This paper addresses the question of dynamic formulation of the six-degrees-of-freedom parallel manipulator known as the Stewart platform. Dynamic equations for the two widely used kinematic structures of the Stewart platform manipulator are derived in closed form through the Newton–Euler approach. The Newton–Euler approach, which is mostly used for inverse dynamics alone in the case of serial manipulators is seen to have an advantage in the case of parallel manipulators for the derivation of closed-form dynamic equations as well. The dynamic equations derived are implemented for forward dynamics of the Stewart platform and some simulation results are also presented.

Development of efficient closed-form dynamic equations for robot manipulators using parallel and perpendicular concepts

This paper presents a novel method that uses parallel and perpendicular concepts to develop the efficient closed-form dynamic equations for robot manipulators. The proposed method applies to any rigid-link robot manipulator with rotational and/or translational joints. The high computational efficiency of the method can make the development of the closed-form dynamic equation for robot manipulators to be implemented on a current microcomputer in real-time. For a general industrial robot manipulator with six degrees-of-freedom, the method needs at most 568 multiplications and 582 additions for this development including the computation of all the dynamics coefficients and actuaor forces/torques of the robot manipulator. For a practical six-link manipulator, the number of mathematical operations of the method is certainly much less than this due to the fact that many parallel and/or perpendicular relationships exist between axes of the distinct link coordinate systems. As generally compared with other existing method for the above purpose, the method proposed in this paper has significantly higher computational efficiency and is computationally the fastest of all the existing methods.

The application of newton‐euler recursive methods to the derivation of closed form dynamic equations

AbstractThe article presents, in tutorial format, a development of the Newton‐Euler (NE) approach to the analysis of robot dynamics. Beginning with fundamental concepts drawn from vector calculus and mechanics, a set of recursive equations are developed which allow the calculation of the dynamics of a manipulator in closed form. An example based on conventional manipulator design is evaluated in some detail and shows that, for kinematic models of some complexity, the NE approach is as fast as other techniques based on Lagrangian methods for deriving closed form dynamical equations.