Forward Dynamics Problem Research Articles

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

Analytical differentiation of the articulated-body algorithm: a geometric multilinear approach

This paper presents a new geometric algorithm to efficiently compute the analytical differentiation of the Articulated-body Algorithm (ABA) with respect to the state variables. Despite the fact that ABA solves the forward-dynamics problem in linear time for branched-multibody systems, its explicit differentiation is not straightforward, and computationally demanding tensor products and contractions appear. The proposed algorithm makes use of Lie algebraic and multilinear operations to recursively exploit the underlying sparsity of the linearization problem. As a result, the arithmetic complexity is dramatically reduced without affecting the analytical solution. Since the linealization of forward dynamics is of great importance in robot-trajectory optimization, a differential dynamic programming solver is employed to demonstrate the performance of our algorithm with humanoid robot models such as NAO and HRP-2. In addition, we provide the computational cost with different robots using an optimized C++ implementation that can be found at https://github.com/garechav/geombd_crtp.git .

Simulation evaluation of dynamic characteristic identification accuracy of sliding bearing considering test error

Aiming at the dynamic characteristics test bench of sliding bearings, the dynamic model is established. Based on the forward and inverse dynamic problems of the bearing, a simulation evaluation method for the identification accuracy of the sliding bearing dynamic characteristics is proposed and the algorithm is verified. The identification errors of dynamic characteristic coefficients under different excitation frequencies are analyzed, the sensitivities of single frequency excitation method and dual-frequency excitation method to test error are contrastively analyzed, and the influence laws of dynamic characteristic identification accuracy of sliding bearing are evaluated. Based on which the traditional single frequency excitation method has been improved. The dynamic characteristic test should be carried out respectively in the low frequency range and the high frequency range. The main stiffness and cross damping are the average of two tests, the main damping is the identification value in the high frequency, and the cross stiffness is the identification value in the low frequency. That will effectively reduce the impact of test error. The obtained data and laws could support the improvement of the dynamic characteristics test method of sliding bearings and the confirmation of test parameters, thereby the accuracy of dynamic characteristics identification is improved.

Friction-Inclusive Modeling of Sliding Contact Transmission Systems in Robotics

This article presents a unified mathematical approach for modeling, identifying, and solving the friction-inclusive dynamics of robotic mechanisms driven by sliding contact transmission systems. This approach is applied to the most common industrial screw-based drives: 1) the worm drive, 2) the simple lead screw drive, and 3) the antibacklash lead screw drive. The resulting dynamics, handily transferable to single and multiple degree of freedom (DoF) manipulators, are complemented with an algorithm for the solution of the forward dynamics problem as well as a framework for friction identification based on nonlinear optimization. The presented theory is experimentally tested on single-DoF screw-based drives for a variety of payloads and at different operation speeds. Simulation and parameter identification results using the classical Coulomb model, the Coulomb model with Stribeck friction, and the Armstrong model with Stribeck friction, rising static friction, and frictional memory are compared with experimental data, showcasing the effectiveness of the presented approach toward framing the concepts of friction and motion transmission into a robotics setting.

Design, Analysis and Simulation of a 6-DOF Serial Manipulator

Within the scope of this study, firstly, mechanical structure of a six-axis serial manipulator was designed. Forward kinematic analysis was made using the Denavit-Hartenberg method, which provides the transition between cartesian coordinates, that is, the linear and angular positions of the end-effector, and joint coordinates; joint angles and linear displacements. In addition, to obtain the relation between the speed of joint variables and the speed of end effector, Jacobian matrix was derived. Dynamic analysis of the system based on Lagrange-Euler mathematical model was acquired. After the design phase was carried out in three-dimensional environment, the physical system-based dynamic model was obtained by transferring this data to the MATLAB-Simscape environment. Inverse dynamic problem was solved to verificate the 3D design of robot and suitablility of selected motors. To solve this problem, position, velocity and acceleration trajectories was given to dynamic model. As a result of this, each joint torques were obtained. The trajectories used in inverse dynamics were calculated using a fifth order polynomial function. Afterwards, in order to test the operation of the system in a simulated environment, PID based controller structures were applied to the dynamic model in MATLAB/Simulink simulation envirenmont and forward dynamic problem was reviewed and discussed.

A procedure to estimate normal and friction contact parameters in the stance phase of the human gait

A new approach to estimate normal and tangential contact parameters in the foot-ground contact during human gait was proposed. A correct estimation of the contact parameters would be very important in the resolution of predictive forward dynamic problems. The normal contact forces have been well estimated in the literature. But accurate estimation of tangential forces has not been reached yet. This work proposed a new procedure to accurately estimate friction forces. The approach has been based on the consideration of the modulus of the tangential force instead of its components. This modulus was introduced together with the modulus of the normal contact force and its two associated moments in an optimization algorithm to fit the contact forces provided by the model to the experimental data obtained with a force plate. An inverse dynamics problem was solved as a step previous to the optimization algorithm. The results showed that both the normal and tangential forces and the moments in the horizontal plane were in agreement with the experimental measurements. This work also analyzed the influence on the results of the friction law. The results obtained with the general friction law, which considered dry (static and dynamic) and viscous friction, were compared with results provided by simpler laws. The analysis of the components of the friction forces pointed out the importance of the Stribeck component in the resultant force instead of the viscous friction which played a minimal role. But for modelling the stick-slip transition, the implementation of a general friction law is necessary.

Forward and Inverse Dynamics of a Unicycle-Like Mobile Robot

In this research work, a new method for solving forward and inverse dynamic problems of mechanical systems having an underactuated structure and subjected to holonomic and/or nonholonomic constraints is developed. The method devised in this paper is based on the combination of the Udwadia-Kalaba Equations with the Underactuation Equivalence Principle. First, an analytical method based on the Udwadia-Kalaba Equations is employed in the paper for handling dynamic and control problems of nonlinear nonholonomic mechanical systems in the same computational framework. Subsequently, the Underactuation Equivalence Principle is used for extending the capabilities of the Udwadia-Kalaba Equations from fully actuated mechanical systems to underactuated mechanical systems. The Underactuation Equivalence Principle represents an efficient method recently developed in the field of classical mechanics. The Underactuation Equivalence Principle is used in this paper for mathematically formalizing the underactuation property of a mechanical system considering a particular set of nonholonomic algebraic constraints defined at the acceleration level. On the other hand, in this study, the Udwadia-Kalaba Equations are analytically reformulated in a mathematical form suitable for treating inverse dynamic problems. By doing so, the Udwadia-Kalaba Equations are employed in conjunction with the Underactuation Equivalence Principle for developing a nonlinear control method based on an inverse dynamic approach. As shown in detail in this investigation, the proposed method can be used for analytically solving in an explicit manner the forward and inverse dynamic problems of several nonholonomic mechanical systems. In particular, the tracking control of the unicycle-like mobile robot is considered in this investigation as a benchmark example. Numerical experiments on the dynamic model of the unicycle-like mobile robot confirm the effectiveness of the nonlinear dynamic and control approaches developed in this work.

Dynamic behavior of a hydraulic crane operating a freely suspended payload

We describe an investigation of the dynamic behavior of a hydraulically driven crane with a freely suspended payload during luffing and slewing motions. To simplify the task, the two movements are considered separately. Taking into account only one motion at a time, the crane is regarded as a three-link kinematic chain with revolute joints. The forward dynamics problem is solved for a crane with three rotational degrees of freedom, two of which describe the load swinging. In both the cases studied, the links are driven by a torque applied via a hydraulic drive, i.e., a linear actuator for the luffing case and a rack and pinion mechanism for the slewing motion. To compose the set of differential equations for the forward dynamics problem, a method based on a general Newton-Euler algorithm is used. From these simulations the time histories of various parameters, namely the swinging angles, hydraulic pressures, and joint forces, are determined. The results obtained via simulations are confirmed experimentally and a good agreement between the two outputs is observed. The results also show that a hydraulic drive system using fast opening flow direction control valves increases the load swing and imposes extensive inertial forces and problems of fatigue and reliability.

Estimation of muscle activity using higher-order derivatives, static optimization, and forward-inverse dynamics

We propose a new method to estimate muscle activity in a straightforward manner with high accuracy and relatively small computational costs by using the external input of the joint angle and its first to fourth derivatives with respect to time. The method solves the inverse dynamics problem of the skeletal system, the forward dynamics problem of the muscular system, and the load-sharing problem of muscles as a static optimization of neural excitation signals. The external input including the higher-order derivatives is required for a calculation of constraints imposed on the load-sharing problem. The feasibility of the method is demonstrated by the simulation of a simple musculoskeletal model with a single joint. Moreover, the influences of the muscular dynamics, and the higher-order derivatives on the estimation of the muscle activity are demonstrated, showing the results when the time constants of the activation dynamics are very small, and the third and fourth derivatives of the external input are ignored, respectively. It is concluded that the method can have the potential to improve estimation accuracy of muscle activity of highly dynamic motions.

Dynamic Modeling and Analysis of a Lightweight Robotic Manipulator in Joint Space

Abstract The primary importance of the paper is the application of the efficient formulation for the simulation of open-loop lightweight robotic manipulator. The framework employed in the paper makes use of the spatial operator algebra and the associated equations are expressed in joint space. This compact representation of the manipulator dynamics makes it possible to solve the robot forward and inverse dynamics problems in a recursive and fast manner. In the current form, the presented algorithm can be applied for the dynamics simulation of an open-loop chain system possessing any number of joints. Specifically, the formulation has been successfully applied for the analysis of the 7DOF KUKA LWR robot. Results from a number of test cases for the robot demonstrate the verification of the calculations

A novel approach for forward dynamic analysis of 3-PRS parallel manipulator with consideration of friction effect

In this paper, a novel decomposition approach to formulate the dynamic model of a 3-Prismatic, Revolute, Spherical (3-PRS) parallel manipulator is proposed. Since the constraint forces arising from the holonomic constraints would not generate a net force or torque to manipulate the motion of the moving platform, the fact motivates to decompose the reaction forces applied to the connecting joints. The decomposition leads to a transformation matrix which can project the dynamic forces of the moving platform formulated in the task space into the joint space. A sufficient condition is determined to guarantee the existence of the projection matrix . Based on the proposed approach, the inverse of 21×21 augmented matrix formulated by conventional approach in solving forward dynamic problem can be decomposed into that of 6×6 and 15×15 matrices. The computational efficiency can therefore be improved about 23.5%. Besides, since the reaction forces can be calculated simultaneously, each of the actuated legs can be decoupled to calculate the normal forces applied to the sliding planes of the prismatic joints . This feature makes it possible to include the effect of corresponding friction forces into consideration. Since the normal forces applied to the sliding planes vary with the reaction forces, the corresponding friction forces are not only the function of sliding velocities but also the dynamics of the whole mechanism. Computer simulations are performed to verify the proposed approach and analyze the effect of the friction forces on the motion accuracy of the moving platform. • We propose a novel decomposition approach to formulate the dynamic model of a 3-PRS (Prismatic, Revolute, Spherical) parallel manipulator. The dynamic model can be utilized to simulate the motion trajectory of the moving platform once the externally applied forces are given. • The decomposition approach leads to a transformation matrix which can project the dynamic forces of the moving platform formulated in the task space into the joint space. • The inverse of 21×21 augmented matrix formulated by conventional approach in solving forward dynamic problem can be decomposed into that of 6×6 and 15×15 matrices. The computational efficiency can therefore be improved about 23.5%. • Each of the actuated legs can be decoupled to calculate the normal forces applied to the sliding planes of the prismatic joints. • The effect of corresponding friction forces is included into consideration.

A Hamiltonian conserving indirect optimal control method for multibody dynamics

AbstractIn the past, a lot of effort has gone into the development of structure‐preserving time‐stepping schemes for forward dynamic problems. This is due to the superior numerical stability of these integrators. Guided by previous developments in the design of energy‐momentum integrators for forward dynamic problems, a Hamiltonian conserving indirect optimal control method will be introduced. For the state equations, a consistent variant of the midpoint evaluation introduced in [1] will be applied. Based on this specific discretization of the state equations, a discretization of the costate equations will be introduced, which is based on the notion of a discrete derivative and which leads to the algorithmic conservation of the discrete Hamiltonian. The newly developed method will be compared with a direct transcription method. We will test the newly proposed method within two numerical examples, which are the optimal control of a particle in a gravitational field and a 3‐link manipulator. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A modular and efficient approach to computational modeling and sensitivity analysis of robot and human motion dynamics

AbstractIn this paper a new class library for the computation of the forward multi‐body‐system (MBS) dynamics of robots and biomechanical models of human motion is presented. By the developed modular modeling approach the library can be flexibly extended by specific modeling elements like joints with specific geometry or different muscle models and thus can be applied efficiently for a number of dynamic simulation and optimization problems. The library not only provides several methods for solving the forward dynamics problem (like articulated body or composite rigid body algorithms) which can transparently be exchanged. Moreover, the numerical solution of optimal control problems, like in the forward dynamics optimization of human motion, is significantly facilitated by the computation of the sensitivity matrix with respect to the control variables. Examples are given to demonstrate the efficiency of the approach. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Modeling and motion simulation of a three-wheeled mobile robot with front wheel driven and steered taking into account wheels’ slip

The problem of modeling of dynamics of a three-wheeled mobile robot with front wheel driven and steered is analyzed in this paper. Kinematical structure and kinematics of the robot are described. A universal methodology of analytical modeling of robot’s dynamics is applied. This methodology takes into account wheel-ground contact conditions and wheels’ slip. Its essence is the use of a contact model of deformable tire with rigid ground and division of the robot’s dynamics model into parts connected with wheels, including tire model, and with the mobile platform. The tire model used in this paper results from empirical dependencies determined during investigations of car tires. Ground geometry and type are specified in the environment model. Tire-ground interface is characterized by coefficients of friction and rolling resistance. The robot model takes into account the presence of friction in kinematical pairs. The model of servomotors is included as well. The important part of this work is simulation research performed using Matlab/Simulink package. Simulation research includes solving of the forward and inverse dynamics problems as well as the tracking control task. During simulations, the robot was moving on concrete and on a piece of ice. The simulation research enabled verification of the elaborated solutions.

Solution of the Forward Dynamics of a Single-loop Linkage Using Power Series

The kinematic differential equations express the paths taken by points, lines, and coordinate frames attached to a rigid body in terms of the instantaneous screw for the motion of that body. Such differential equations are linear but with a time-varying coefficient and hence solvable by power series. A single-loop kinematic chain may be expressed by a system of such differential equations subject to a linear constraint. A single matrix factorization followed by a sequence of substitutions of linear-system right-hand-side terms determines successive orders of the joint rate coefficients in the kinematic solution for this mechanism. The present work extends this procedure to the forward dynamics problem, applying it to a Clemens constant-velocity coupling expressed as a spatial 9R closed kinematic chain.

Implicit methods for efficient musculoskeletal simulation and optimal control

The ordinary differential equations for musculoskeletal dynamics are often numerically stiff and highly nonlinear. Consequently, simulations require small time steps, and optimal control problems are slow to solve and have poor convergence. In this paper, we present an implicit formulation of musculoskeletal dynamics, which leads to new numerical methods for simulation and optimal control, with the expectation that we can mitigate some of these problems. A first order Rosenbrock method was developed for solving forward dynamic problems using the implicit formulation. It was used to perform real-time dynamic simulation of a complex shoulder arm system with extreme dynamic stiffness. Simulations had an RMS error of only 0.11 degrees in joint angles when running at real-time speed. For optimal control of musculoskeletal systems, a direct collocation method was developed for implicitly formulated models. The method was applied to predict gait with a prosthetic foot and ankle. Solutions were obtained in well under one hour of computation time and demonstrated how patients may adapt their gait to compensate for limitations of a specific prosthetic limb design. The optimal control method was also applied to a state estimation problem in sports biomechanics, where forces during skiing were estimated from noisy and incomplete kinematic data. Using a full musculoskeletal dynamics model for state estimation had the additional advantage that forward dynamic simulations, could be done with the same implicitly formulated model to simulate injuries and perturbation responses. While these methods are powerful and allow solution of previously intractable problems, there are still considerable numerical challenges, especially related to the convergence of gradient-based solvers.

A divide-and-conquer direct differentiation approach for multibody system sensitivity analysis

In the design and analysis of multibody dynamics systems, sensitivity analysis is a critical tool for good design decisions. Unless efficient algorithms are used, sensitivity analysis can be computationally expensive, and hence, limited in its efficacy. Traditional direct differentiation methods can be computationally expensive with complexity as large as O(n4+n2m2+nm3), where n is the number of generalized coordinates in the system and m is the number of algebraic constraints. In this paper, a direct differentiation divide-and-conquer approach is presented for efficient sensitivity analysis of multibody systems with general topologies. This approach uses a binary tree structure to traverse the topology of the system and recursively generate the sensitivity data in linear and logarithmic complexities for serial and parallel implementations, respectively. This method works concurrently with the forward dynamics problem, and hence, requires minimal data storage. The differentiation required in this algorithm is minimum as compared to traditional methods, and is generated locally on each body as a preprocessing step. The method provides sensitivity values accurately up to integration tolerance and is insensitive to perturbations in design parameter values. This approach is a good alternative to existing methodologies, as it is fairly simple to implement for general topologies and is computationally efficient.

DYNAMIC MODELING OF FOUR-WHEEL FULL-SUSPENSION VEHICLE

This paper presents an analytical dynamic model of a four-wheel full-suspension vehicle using 24 independent dof. The methodology presented herein is capable of simulating vehicle dynamics on any terrain so long as it is smooth enough that tires can remain in contact. Given the steering torque, driving torque and terrain structure, the forward dynamics problem is solved to obtain velocities, accelerations, forces and torques of all joints. Among these, contact forces between tires and the road play a crucial role in designing an early sensory feedback system of imminent vehicle roll-over.

DYNAMIC SIMULATION OF A SPATIAL 3-DOF TENSEGRITY MECHANISM

Tensegrity mechanisms have the advantage of being relatively lightweight due to. their extensive use of cables and springs. As such, they have the potential of being an attractive alternative to conventional mechanisms in certain application environments. However, the presence of unconstrained degrees of freedom in tensegrity mechanisms leads to a dynamic behaviour that cannot be directly controlled with the actuators. In this work, the dynamic model of a novel spatial three-degree-of-freedom (3-DOF) tensegrity mechanism is developed using the Lagrangian formulation. The resulting equations of motion are then solved to simulate the mechanism's motion between equilibrium configurations. Since the mechanism is subjected to holonomic nonlinear geometrical constraints, these must be considered during the solution of its forward dynamic problem. It is seen that the use of damping in the springs is not very efficient in dissipating the mechanism's energy during motion.

Forward Dynamics of Two-Dimensional Skeletal Models. A Newton-Euler Approach

Mathematical modeling and computer simulation play an increasingly important role in the search for answers to questions that cannot be addressed experimentally. One of the biggest challenges in forward simulation of the movements of the musculoskeletal system is finding an optimal control strategy. It is not uncommon for this type of optimization problem that the segment dynamics need to be calculated millions of times. In addition, these calculations typically consume a large part of the CPU time during forward movement simulations. As numerous human movements are two-dimensional (2-D) to a reasonable approximation, it is extremely convenient to have a dedicated, computational efficient method for 2-D movements. In this paper we shall present such a method. The main goal is to show that a systematic approach can be adopted which allows for both automatic formulation and solution of the equations of kinematics and dynamics, and to provide some fundamental insight in the mechanical theory behind forward dynamics problems in general. To illustrate matters, we provide for download an example implementation of the main segment dynamics algorithm, as well as a complete implementation of a model of human sprint cycling.