Forward Dynamic Equations Research Articles

Forward Dynamics Estimation from Data-Driven Inverse Dynamics Learning??

In this paper, we propose to estimate the forward dynamics equations of mechanical systems by learning a model of the inverse dynamics and estimating individual dynamics components from it. We revisit the classical formulation of rigid body dynamics in order to extrapolate the physical dynamical components, such as inertial and gravitational components, from an inverse dynamics model. After estimating the dynamical components, the forward dynamics can be computed in closed form as a function of the learned inverse dynamics. We tested the proposed method with several machine learning models based on Gaussian Process Regression and compared them with the standard approach of learning the forward dynamics directly. Results on two simulated robotic manipulators, a PANDA Franka Emika and a UR10, show the effectiveness of the proposed method in learning the forward dynamics, both in terms of accuracy as well as in opening the possibility of using more structured models.

Deflection estimation of industrial robots with flexible joints

The application of industrial robots in manufacturing industries has received considerable concerns due to the high flexibility, multifunctionality, and cost-efficiency. It is well known that the robot positioning accuracy is susceptible to the load and motion of robots owing to the insufficient stiffness of robots. Therefore, the machining accuracy improvement has been a research focus in the robotic manufacturing industries in the last decade. To overcome the measurement difficulty of the joint torque and position as well as the complex dynamic coupling between rotors and links, two forward dynamics algorithms for the robot deflection estimation are proposed in this paper. The robot kinematics and dynamics algorithms considering the dynamic coupling between rotors and links are developed based on Lie theory. The forward dynamics equations of robots are solved via the proposed algorithms: the implicit numerical integration algorithm and numerical iterative estimation algorithm. When only the motor position is available, the implicit numerical integration algorithm is employed to solve the forward dynamics equations to estimate the joint torque and position. At the same time, when both the motor position and torque are available, the forward dynamics equations can be reorganized as algebraic equations and solved by the numerical iterative estimation algorithm. Simulations of a 6-DOF serial robot are performed to verify the accuracy of the proposed algorithms.

Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism.

The solution of the dynamic equations of the six-axis accelerometer is a prerequisite for sensor calibration, structural optimization, and practical application. However, the forward dynamic equations (FDEs) and inverse dynamic equations (IDEs) of this type of system have not been completely solved due to the strongly nonlinear coupling relationship between the inputs and outputs. This article presents a comprehensive study of the FDEs and IDEs of the six-axis accelerometer based on a parallel mechanism. Firstly, two sets of dynamic equations of the sensor are constructed based on the Newton–Euler method in the configuration space. Secondly, based on the analytical solution of the sensor branch chain length, the coordination equation between the output signals of the branch chain is constructed. The FDEs of the sensor are established by combining the coordination equations and two sets of dynamic equations. Furthermore, by introducing generalized momentum and Hamiltonian function and using Legendre transformation, the vibration differential equations (VDEs) of the sensor are derived. The VDEs and Newton–Euler equations constitute the IDEs of the system. Finally, the explicit recursive algorithm for solving the quaternion in the equation is given in the phase space. Then the IDEs are solved by substituting the quaternion into the dynamic equations in the configuration space. The predicted numerical results of the established FDEs and IDEs are verified by comparing with virtual and actual experimental data. The actual experiment shows that the relative errors of the FDEs and the IDEs constructed in this article are 2.21% and 7.65%, respectively. This research provides a new strategy for further improving the practicability of the six-axis accelerometer.

Controllability of Nash Equilibrium in Game-Based Control Systems

Controlling complex systems to desired states is of primary importance in science and engineering. In the classical control framework, the plants to be controlled usually do not have their own payoff or objective functions; however, this is not the case in many practical situations in, for examples, social, economic, and “intelligent” engineering systems. This motivates our introduction of the game-based control system (GBCS), which has a hierarchical decision-making structure: one regulator and multiple agents. The regulator is regarded as the global controller that makes decision first, and then, the agents try to optimize their respective objective functions to reach a possible Nash equilibrium as a result of noncooperative dynamic game. A fundamental issue in the GBCS is: Is it possible for the regulator to change the macrostates by regulating the Nash equilibrium formed by the agents at the lower level? This leads to the investigation of controllability of the Nash equilibrium of the GBCS. In this paper, we will first formulate this new problem in a general nonlinear framework and then focus on linear systems. Some explicit necessary and sufficient algebraic conditions on the controllability of the Nash equilibrium are given for a linear GBCS, by solving the controllability problem of the associated forward and backward dynamic equations, which is a key technical issue and has rarely been explored in the literature.

Dynamic modeling and experimental validation of skid-steered wheeled vehicles with low-pressure pneumatic tires on soft terrain

Wheeled skid-steered technology has an increasing interest in its use for off-road unmanned ground vehicles, because of its great mobility and compact mechanical structure. By integrating multibody dynamics model and semi-empirical tire-terrain model, this paper presents a dynamic modeling approach for skid-steered wheeled vehicles with low-pressure pneumatic tires on soft terrain to predict and investigate its steering performance. The forward dynamics equations are built by spatial vector algebra. The tire–terrain model estimates flexible deformation and sinkage of the tire, and calculates forces and torques exerted on the tire according to relative motions of tire–terrain contact. The combined longitudinal slip and lateral skid of tires, and the vertical coupled deformations of tires and terrain are also considered in tire-terrain model. This approach optimizes the solution procedure and improves the computing efficiency. The simulation results show that the proposed tire–terrain model can predict the effects of rigid and flexible operation modes of tires on mechanical properties of tires and steering performances of the vehicle. The proposed dynamic model is validated on a six-wheeled skid-steered vehicle. The comparisons between experimental results and simulations show that the proposed dynamic model provides a better accuracy of steering performance simulation for skid-steered vehicles.

Planar Multibranch Open-Loop Robotic Manipulators Subjected to Ground Collision

In this article, a recursive approach is used to dynamically model a tree-type robotic system with floating base. Two solution procedures are developed to obtain the time responses of the mentioned system. A set of highly nonlinear differential equations is employed to obtain the dynamic behavior of the system when it has no contact with the ground or any object in its environment (flying phase); and a set of algebraic equations is exploited when this tree-type robotic system collides with the ground (impact phase). The Gibbs–Appell (G–A) formulation in recursive form and the Newton’s impact law are applied to derive the governing equations of the aforementioned robotic system for the flying and impact phases, respectively. The main goal of this article is a systematic algorithm that is used to divide any tree-type robotic system into a specific number of open kinematic chains and derive the forward dynamic equations of each chain, including its inertia matrix and right-hand side vector. Then, the inertia matrices and the right-hand side vectors of all these chains are automatically integrated to construct the global inertia matrix and the global right-hand side vector of the whole system. In fact, this work is an extension of Shafei and Shafei (2016, “A Systematic Method for the Hybrid Dynamic Modeling of Open Kinematic Chains Confined in a Closed Environment,” Multibody Syst. Dyn., 38(1), pp. 21–42.), which was restricted to a single open kinematic chain. So, to show the effectiveness of the suggested algorithm in deriving the motion equations of multichain robotic systems, a ten-link tree-type robotic system with floating base is simulated.

Logarithmic complexity dynamics formulation for computed torque control of articulated multibody systems

Divide-and-Conquer Algorithm is extended and integrated with the method of Computed Torque Control (CTC) to model and control fully-actuated multibody systems. CTC uses inverse and forward dynamics to control a multibody problem. This may impose unnecessary computational cost on the simulation if the corresponding equations are not formed and solved wisely. Additionally, this technique mainly uses the dynamic equations formulated based on the minimum number of joint coordinates. Herein, the Generalized Divide-and-Conquer Algorithm (GDCA) with the capability of accommodating both spatial and generalized forces is used to efficiently form the forward dynamic equations. Furthermore, a new mathematical formulation is generated to adjust the inverse-GDCA containing a redundant set of spatial and generalized coordinates with the joint-coordinate-based inverse dynamics utilized in CTC. Both GDCA and iGDCA apply a series of assembly and disassembly passes to form and solve the forward and inverse dynamic equations without generating the mass and Jacobian matrices of the entire system. This paper also addresses the challenges which may appear in controlling fully-actuated multibody systems with kinematic loops. The GDCA-based CTC is then equipped with popular control techniques such as PID, pole-placement-based state feedback, and linear quadratic regulator, and used to control selected multibody systems.

General Dynamic Model for Skid-Steer Mobile Manipulators With Wheel–Ground Interactions

The design of motion controllers that could reduce terrain disturbances propagated to the arm and optimize the trajectories of skid-steer mobile manipulators (SSMMs) working in agricultural, construction, or mining environments requires accurate physical models capable of correctly representing disturbances caused by changes in terrain slope, loss of traction, and other wheel-ground interactions. Therefore, this paper develops the forward dynamics equations of motion and proposes a general model for SSMMs. The novelty of the model is that it considers a floating-base with nonpermanent wheel-ground contacts, i.e., the mobile base has 6-degree of freedom (DOF) and the wheels may lose contact with the terrain. Unlike the existing models, the proposed one is general because the procedure can be extended to MMs with an arbitrary number of arms or wheels and does not restrict the mobile base to planar 2-D motions. Furthermore, it takes into account the loss of traction and the deformation of the wheel–ground contacts. The experimental results involving a small SSMM with a 3-DOF arm and an industrial skid-steer loader validate the physical fidelity of the model. Although the focus of this paper is not in control aspects, one of the examples shows the role physically accurate models can have in the design of end-effector motion controllers, and its potential application to the design of controllers capable of decoupling the arm motion from ground disturbances.

Asymptotic analysis of Painlevé’s paradox

Painlevé’s paradox is a well-known problem in rigid-body dynamics, of which the forward dynamics equations could have no solution. To handle this situation, an assumption of tangential impact is often introduced. Although the assumption seems to provide a good fix, it still needs to be mathematically examined via analyzing the asymptotic property of a compliance-based model at the limit of rigidity. In this paper, we revisit the paradox using the typical Painlevé’s example of a rod sliding on a rough surface. For convenience, the interaction at the contact point of the rod is represented by a linear spring to scale the local normal compliance, coupled with Coulomb’s law to reflect friction. We perform an asymptotic analysis using the spring stiffness as a perturbation parameter. The rod dynamics in the Painlevé’s paradox, accompanying the variation of friction status, consists of three distinct phases as follows: An initial period of sliding which allows contact force to diverge with the increase of the spring stiffness, a period of sticking which ends at the occurrence of a reverse slip motion, and a reverse slip phase which causes the rod to be detached from the contact surface. As the stiffness goes to infinity, all the time intervals of the three phases converge to zero. This analysis theoretically confirms the assumption of the tangential impact in the paradox of sliding rod dynamics.

Determination of body segment masses and centers of mass using a force plate method in individuals of different morphology

Body segment masses and center of mass (COM) locations are required to calculate intersegmental forces and net joint moments using inverse or forward dynamics equations. These inertial properties are estimated from methods involving cadavers or living individuals. The present clinical methods are limited to similar populations from which the anthropometric measures were obtained. This study presented a simple force plate method that can be used to determine subject-specific segment masses and COM locations and compared it to other well-known methods. The proposed method was tested in individuals with different body mass index (i.e., lean, normal, and obese) to verify its sensitivity. All the segmental mass and COM values obtained from the force plate method were within the range of those of the other methods for the entire sample. Significant differences were identified between the morphological groups in relative segmental masses at the upper arm and leg and foot, and COM locations at the leg and foot and head and trunk as obtained from the force plate method ( p < 0.05). The proposed method involves direct procedures to determine subject-specific segmental masses and COM locations. It is sensitive to detect differences between various morphological populations.

Dynamics analysis of the Gough-Stewart platform manipulator

A novel derivation of the forward dynamic equations for the Gough-Stewart platform manipulator based on Kane's equation is proposed. In this method, each leg of the Gough-Stewart platform manipulator is treated as an independent substructure, the system dynamic equations are composed of the equations of legs and platform according to the constraints among substructures. The formulation has been implemented in MATLAB routines, and simulation results have been given to show the validation of the new approach. Compared with the traditional Newton-Euler method and Lagrange formulation, the modeling process proposed in the paper is more straightforward and systematic, and the final dynamic equations are very concise.

Stability of frictional contact in constrained rigid-body dynamics

The use of rigid-body models during frictional contact is often justified by proving the existence of a unique solution to the forward dynamic equations. The implicit assumption here is that the contact forces so obtained are stable. In this paper, the rigid-body assumption is relaxed and body-to-body contacts are modeled using springs and dampers. A singular perturbation analysis reveals additional necessary conditions to ensure contact force stability in the reduced rigid-body model. Furthermore, the analysis indicates that stability depends on a damping ratio associated with the rigid contacts.

An original O(N) formulation for simulation of robotic systems

In the last decade a significant number of papers have been published on the efficient formulation of the inverse and forward dynamic equations of motion for robotic manipulators and in general for multibody systems, in view of real-time control or real-time simulation of the system. Until recently these dynamic formulations have been confined to serial computers. The breakthrough of parallel computers however causes to reconsider dynamical formulations from this new perspective. This paper presents an original order N formulation for simulation of the dynamics of a robotic manipulator without inversion of the massmatrix. The proposed algorithm differs from the well-known recursive O( N ) methods. Instead of a backward and two forward recursions, this method requires only one forward recursion and the solution of a small linear system of algebraic equations to generate the equations of motion for open-loop systems. The computations within this forward recursion loop can easily be arranged for parallel processing resulting in a very low computational load. The proposed method can be used for rigid multibody systems in tree structure with arbitrary types of joints. This algorithm also proves to be very promising in case of mechanical systems with closed loops.

The Effect of Friction on the Forward Dynamics Problem

This article discusses the numerical solution of the forward dynamic equations of an n-degree-of-freedom manipulator with friction. Also discussed are the modeling and experimen tal identification of friction. It is shown that the inclusion of Coulomb-type friction in the dynamic equations introduces two difficulties in the forward dynamic solution. The differential equations are shown to be discontinuous in the highest-order derivative terms. In addition, the load dependency of this type of friction typically causes the equations to be implicit in the joint accelerations. For the important case of load-dependent transmission friction, the equations can be explicit. Techniques for the forward solution are described through the example of a roller screw transmission. Experimental and simulation results are used to show the importance of load-dependent friction in a particular robot.

Computational requirements for a discrete Kalman filter in robot dynamics algorithms

SUMMARYIn standard classical kinematic and dynamic considerations the equations of motion for an n-link manipulator can be obtained as recursive Newton-Euler equations. Another approach to finding the inverse dynamics equations is to formulate the system dynamics and kinematics as a two-point boundary-value problem. The equivalence between these two approaches has been proved in this paper. Solution to the two-point boundary-value problem leads to the forward dynamics equations which are similar to the equations of Kalman filtering and Bryson-Frazier fixed time-interval smoothing. The extensive numerical studies conducted by the author on the new inverse and forward dynamics algorithms derived from the two-point boundary-value problem establish the same level of confidence as exists for current methods. In order to obtain the algorithms with the smallest coefficients of the polynomial of order O(n), the categorization procedure has been implemented in this work.