Analytical Inverse Solution Research Articles

Kinematic Models and the Performance Level Index of a Picking-and-Placing Hybrid Robot

The mobile platform of the parallel robot designed for picking and placing operations is usually equipped with one or two extra degree(s) of freedom to enable flexible grasping orientations. However, additional motors indicate extra loads for the moving platform, and the total payload performance shrinks. This paper proposes a spatial picking-and-placing manipulator, in which one actuator that is supposed to be installed on the mobile platform is placed far away from the mobile platform. The platform has a large workspace along one direction. The comprehensive analytical inverse and forward kinematic solutions of this robot are derived. The reachable workspace of the parallel manipulator module is then explored. The novel performance level index is designed to normalize the performance index and demonstrate the performance rank for any pose. A mathematical proof is provided for this novel index. The manipulability index is taken as an example to examine the level indicator. A multi-objective optimization is implemented to pursue optimal performance; then, the initial design and optimized results are compared in detail. A sample trajectory is provided to verify the correctness of the kinematic mathematical model of the parallel mechanism.

Analytical and deep learning approaches for solving the inverse kinematic problem of a high degrees of freedom robotic arm

Inverse kinematics is the basis for controlling the motion of robotic manipulators. It defines the required joint variables for the robotic end-effector accurately reach the desired location. Due to the derivation difficulty, computation complexity, singularity problem, and redundancy, analytical Inverse kinematics solutions pose numerous challenges to the operation of many robotic arms, especially for a manipulator with a high degree of freedom. This paper develops different Deep Learning networks for solving the Inverse kinematics problem of six- Degrees of Freedom robotic manipulators. The implemented neural architectures are Artificial Neural Network, Convolutional Neural Network, Long–Short Term Memory, Gated Recurrent Unit, and Bidirectional Long–Short Term Memory. In this context, we associate the proposed results with a specific tuning of Deep Learning network hyper-parameters (number of hidden layers, learning rate, Loss function, optimization algorithm, number of epochs, etc.). The Bidirectional Long–Short Term Memory network outperformed all proposed architectures. To be close as possible to the experimental results, we have included two types of noise in the training data set to validate which of the five proposed neural networks is more efficient. Furthermore, in this study, we compare the performance of analytical and soft computing solutions in generating robots’ trajectories. We include this scenario, focusing on the advantage of implementing neural networks in avoiding the singularity problem that can occur using the analytical approach. In addition, we used the RoboDK simulator to show simulation results with real-world meaning. The performance of Deep Learning models depends on the complexity of the posed problem. Moreover, the complexity of the Inverse Kinematics problem is related to the number of Degrees of Freedom. At the end of this work, we evaluate the influence of the complexity of robotic manipulators on the proposed Deep Learning networks’ performance. The results show that the implemented Deep Learning mechanisms performed well in reaching the desired pose of the end-effector. The proposed inverse kinematics strategies apply to other manipulators with different numbers of Degrees of Freedom.

A real-time explicit mapping and teleoperation control method for humanoid robots with posture constraints

In most industrial robot applications, on-site robot operation is required. However, this is not feasible for certain harsh operating conditions, such as high temperatures and intense radiation. In such instances, teleoperation based on machine vision can be applied, and adding posture constraints helps control the robot to complete more complex tasks. The general idea is to use quadratic programming or other optimization methods to solve the problem. In order to improve accuracy and ensure real-time performance, a real-time explicit mapping and teleoperation control (REMTC) is proposed. Based on the principle of "included angle minimization" (IAM), the incorrect mapping problem which might be caused by the traditional method is solved. In addition, considering that the analytical inverse solution algorithm is not as universal as the general optimization framework, an improved structure transformation method is proposed, so that the algorithm can be applied to both Spherical- Revolute-Spherical (SRS) and non-SRS robots. Finally, the experiment verifies the effectiveness of the algorithm in real-time performance, accuracy and reliability.

Analytical inverse solution of a rotational achromatic Risley prism based on Fourier series fitting

High-precision analytical inverse solution is a significant issue faced by the rotational achromatic Risley prism system. The current inverse solution method is based on numerical iteration, which is time-consuming. To solve this problem, we propose an analytical inverse solution method for a rotational achromatic Risley prism system based on Fourier series fitting. In the proposed method, the expression of the difference between the rotation angle and elevation angle of beam deflection is inversely fitted using the forward solution model. The analytical inverse solution is achieved using a two-step method based on this fitted expression. Simulation results of the analytical inverse solution show that the maximum azimuthal error was less than 2.23e−8 arcsec and the maximum elevation error was less than 1.33e−4 arcsec. Experimental results prove that with 1 arcsec pointing accuracy, the time consumption of the proposed method is only 0.108 ms, whereas that of the previous numerical-iteration-based method is 1.480 ms.

Configuration design and dimensional synthesis of an asymmetry 2R1T parallel mechanism

AbstractThis paper focuses on configuration design, dimensional synthesis, and engineering application of a novel asymmetric 2R1T parallel mechanism (PM) with zero-coupling degree. The analytical forward and inverse displacement solutions are deduced by the means of vector method. The mathematical models between Euler angles and the orientational parameters (i.e., azimuth and tilt angles) of the offset output axis are established. Using screw theory as mathematical tool, this paper worked out evaluation indices of motion/force transmissibility and presented the definitions and calculation methods of good transmission orientation workspace and good transmission orientation capacity (GTOC). Furthermore, a comparative example with respect to kinematic performance of asymmetric UPS-RPU-PU PM and planar symmetric 2UPS-PU PM is carried out, and the result demonstrates that UPS-RPU-PU significantly outperforms 2UPS-PU in terms of GTOC. The constrained optimization model is constructed to formulate optimal problem of dimensional parameters on the maximizing GTOC, which is then solved by differential evolution algorithm. Finally, an engineering case demonstrates that the optimized mechanism has a good application prospect in hydraulic support test bed.

General inverse kinematics method for 7-DOF offset manipulators based on arm angle parameterization

This paper proposes a general numerical methodology of inverse kinematics computation for 7-degree-of-freedom offset manipulators based on arm angle parameterization. First, analytical inverse kinematics solutions for two types of Spherical-Revolute-Spherical manipulators are derived based on arm angle parameterization. Then, the inverse kinematics of the offset manipulator is solved using a simplified manipulator. In each iteration of the algorithm, the inverse solution of the simplified manipulator is used to calculate the position and arm angle error between the actual manipulator and desired value. The error is utilized to compensate the position and arm angle of the simplified manipulator so that the inverse solution of the simplified manipulator gradually approaches the desired solution of the actual manipulator. Moreover, the problems of workspace reduction due to structure simplification and the placement of the iteration point outside of the workspace of the simplified manipulator are optimized. Additionally, the algorithm singularity problem of the offset manipulator is solved via the dual arm angle parameterization method. Finally, the method is validated by numerical simulation and physical experiments.

Applications of an improved Dixon elimination method for the inverse kinematics of 6R manipulators

Variable elimination of high dimensional polynomials is an essential problem for robot inverse kinematics, but it is hard to approach both high efficiency and precision simultaneously. This study proposes a unique elimination method for vector polynomial systems established for kinematic modelling of six-revolute serial manipulators, and then applies it to derive analytical inverse kinematic solutions in real time with high absolute precision. To begin, the kinematic modelling equations are parameterized into a non-redundant vector polynomial system, so as to make full use of vector properties in elimination processing. After that, an improved elimination method based on the Dixon resultant is proposed as a solution to the polynomial computation issue. Since there exist some duplicate terms in the Dixon polynomial, a linear constrained Dixon matrix is created by extracting linear terms from variable sequences. Furthermore, the inverse kinematic problem for six-revolute manipulators is addressed based on the elimination method, wherein the configurations of general decoupling manipulators are designed considering both precise machining and fast solutions. The proposed formulations are ported into the core algorithm of professional software, demonstrating that they are highly accurate and real-time required for precise manipulators of automatic use.

Precise semi-analytical inverse kinematic solution for 7-DOF offset manipulator with arm angle optimization

Seven-degree-of-freedom redundant manipulators with link offset have many advantages, including obvious geometric significance and suitability for configuration control. Their configuration is similar to that of the experimental module manipulator (EMM) in the Chinese Space Station Remote Manipulator System. However, finding the analytical solution of an EMM on the basis of arm angle parameterization is difficult. This study proposes a high-precision, semi-analytical inverse method for EMMs. Firstly, the analytical inverse kinematic solution is established based on joint angle parameterization. Secondly, the analytical inverse kinematic solution for a non-offset spherical-roll-spherical (SRS) redundant manipulator is derived based on arm angle parameterization. The approximate solution of the EMM is calculated in accordance with the relationship between the joint angles of the EMM and the SRS manipulator. Thirdly, the error is corrected using a numerical method through the analytical inverse solution based on joint angle parameterization. After selecting the stride and termination condition, the precise inverse solution is computed for the EMM based on arm angle parameterization. Lastly, case solutions confirm that this method has high precision, and the arm angle parameterization method is superior to the joint angle parameterization method in terms of parameter selection.

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

With the vaccination against Covid-19 now available, how vaccination campaigns influence the mathematical modeling of epidemics is quantitatively explored. In this paper, the standard susceptible-infectious-recovered/removed (SIR) epidemic model is extended to a fourth compartment, V, of vaccinated persons. This extension involves the time t-dependent effective vaccination rate, v(t), that regulates the relationship between susceptible and vaccinated persons. The rate v(t) competes with the usual infection, a(t), and recovery, μ(t), rates in determining the time evolution of epidemics. The occurrence of a pandemic outburst with rising rates of new infections requires k+b<1−2η, where k=μ(0)/a(0) and b=v(0)/a(0) denote the initial values for the ratios of the three rates, respectively, and η≪1 is the initial fraction of infected persons. Exact analytical inverse solutions t(Q) for all relevant quantities Q=[S,I,R,V] of the resulting SIRV model in terms of Lambert functions are derived for the semi-time case with time-independent ratios k and b between the recovery and vaccination rates to the infection rate, respectively. These inverse solutions can be approximated with high accuracy, yielding the explicit time-dependences Q(t) by inverting the Lambert functions. The values of the three parameters k, b and η completely determine the reduced time evolution of the SIRV-quantities Q(τ). The influence of vaccinations on the total cumulative number and the maximum rate of new infections in different countries is calculated by comparing with monitored real time Covid-19 data. The reduction in the final cumulative fraction of infected persons and in the maximum daily rate of new infections is quantitatively determined by using the actual pandemic parameters in different countries. Moreover, a new criterion is developed that decides on the occurrence of future Covid-19 waves in these countries. Apart from in Israel, this can happen in all countries considered.

Singularity-free solutions for inverse kinematics of degenerate mobile robots

This paper describes a method for inverse kinematics that is tolerant to degenerate wheels structures, specifically to control the motion of the transformable, pseudo-omnidirectional mobile platform. This platform can expand its lateral footprint while maintaining the robot’s planar mobility. The proposed method creates an inverse kinematical solution where there previously was none due to modeling singularities from the use of degenerate wheel structures. The idea of degenerate wheel structures is introduced and a robot using only these wheels is described. The first order dynamics of the robot are derived and an analytical inverse kinematical solution to the steering angle and rate of wheel rotation for each wheel is given using closed form solutions. The bifurcating solutions are addressed and as long as the chosen solutions are consistent between the steering angle and the wheel rotation speed, there is no impact on the effectiveness of either pair of solutions. This solution proved to be accurate through simulation of both omnidirectional motion and instantaneous center of rotation-based motion. Consistent results were achieved in experiments.

Self-similarity of surface heat transfer fields on a pre-heated rectangular plate in an obliquely impinging sonic jet

This paper presents an experimental study of the time-dependent surface temperature, heat flux and Nusselt number fields on a pre-heated, rectangular aluminum plate due to an obliquely impinging sonic jet at different total pressures. Surface temperatures are measured by detecting the luminescent emission from temperature sensitive paint. An analytical inverse solution of the one-dimensional time-dependent heat conduction equation is used to determine the heat flux from the temperature measurements. When normalized, the ensemble-averaged heat transfer fields (surface temperature, heat flux and Nusselt number) exhibit self-similarity (time-invariance) for different test conditions and therefore are found to exhibit a somewhat universal nature. A theoretical description of the self-similar behavior is presented for scaling experimental heat transfer data. Furthermore, the skin friction fields extracted from surface temperature and heat flux fields reveal generic topological structures of the flows.

Kinematically Redundant Hybrid Robots With Simple Singularity Conditions and Analytical Inverse Kinematic Solutions

The rotational workspace of a parallel manipulator is mainly limited by type II singularities. In this letter, we present an approach to synthesize three-legged planar $3+n$ and spatial $6+n$ degree-of-freedom kinematically redundant hybrid robots with very simple—or vanishing—type II singularity conditions. Here, $n$ denotes the number of redundancies. Moreover, the inverse kinematic problem of the proposed architectures can be solved analytically. Because of these advantages, such redundant robots are well suited for applications in physical human–robot interaction, where large rotational workspaces are typically needed.

Heat transfer distribution of premixed methane-air laminar flame jets impinging on ribbed surfaces

Heat transfer distribution of premixed methane-air laminar flame jet impinging on ribbed surfaces is presented in this work. Experiments are carried out on ribbed plates with three different geometrical shaped rib elements i.e. circular, rectangular and triangular. In addition, numerical simulations are performed to study flow field distribution near the ribs. During the experiments, Reynolds number is varied from 600 to 1800 and burner tip to target plate distance is varied from 2 to 4. An analytical inverse solution to three dimensional transient heat conduction presented in our previous work is used to obtain heat transfer parameters. Heat transfer coefficients are found lower whereas reference temperatures are observed higher on ribbed surfaces as compared with smooth surface. Obstruction to the flow, flow separation and decrease in momentum are the reasons attributed for lower heat transfer rate for ribbed surfaces.

Analytical forward and inverse solutions using a mathematics model for double wedge prisms in different configurations

In this study, the optical characteristics of double triangle wedge prisms are researched and analyzed, and an innovative mathematics model including four apex angle parameters is proposed for the analytical forward and inverse solutions. A mathematical model that defines the four apex angle parameters for DWPS as applied in different configurations has been demonstrated.

In-situ calibration for temperature-sensitive-paint heat-flux measurement on a finite base

This work incorporates in-situ calibration into the analytical inverse heat transfer solution for a finite base in temperature-sensitive-paint (TSP) measurements in high-enthalpy short-duration hypersonic wind tunnels. The thermal penetration parameter of the polymer layer (TSP) in the analytical inverse solution is determined by in-situ calibration based on the data obtained using a reliable heat flux sensor at a reference location. This method is applied to TSP images obtained on a wedge in a high-enthalpy shock tunnel.

An Inverse Source Problem Connected with Thermoacoustic Imaging in Multi-layer Planar Medium

We derived analytical forward and inverse solution of thermoacoustic wave equation for nonhomogeneous medium. We modelled the nonhomogeneous medium as a multi-layer planar medium and defined initial conditions, continuity conditions on the layer boundaries and radiation conditions at infinity assuming the source distribution existing in all layers. These solutions of thermoacoustic wave equation are based on the method of Green’s functions for layered planar media. For qualitative testing and comparison of the point-spread functions associated with the homogeneous and layered solutions, we performed numerical simulations. Our simulation results showed that the conventional inverse solution based on homogeneous medium assumption, as expected, produced incorrect locations of point sources, whereas our inverse solution involving the multi-layer planar medium produced point sources at the correct source locations. Also, we examined whether the performance of our layered inverse solution is sensitive to medium parameters used as priority information in the measured data. Our inverse solutions based on multi-layer planar media are applicable for cross-sectional two-dimensional imaging of abdominal structure and the organs such as breast and skin.

Analytic inverse solutions for Risley prisms in four different configurations for positing and tracking systems.

The conventional ray deviation formula and analytical inverse solution for Risley prisms are traditionally derived by the first-order paraxial method, which is simplified and does not provide sufficient accuracy. When a light ray enters the wedge prism, the component that is perpendicular to the base side is deviated by the wedge prism and the component parallel to the base side is uninfluenced. In this study, the problem is presented and analyzed from the scalar form of Snell's law, and 2D vector algebra is used to solve the problem. The rotation and refracting angles of a single wedge prism are obtained by analyzing the propagation through the prism of incident rays from different directions, and ray deviation formulas are derived. Four configurations are described with ray deviation formulas and analytical inverse solutions deduced from the corresponding formulas for a single wedge prism. In the final section, two examples are considered for the four configurations. The numeric results demonstrate the accuracy of the inverse solution provided by the proposed method.

Analytical Inverse Solution of Eigenstrains and Residual Fields in Autofrettaged Thick-Walled Tubes

The smoothed inverse eigenstrain method is revisited for the reconstruction of residual fields and eigenstrains from limited strain measurements within axially symmetric tubes. The application of the present approach is successfully demonstrated for two cases of analytical solution and experimental measurements. The well-known advantage of the smoothed inverse eigenstrain approach is that it not only minimizes the deviation of measurements from the model predictions but also will result in an inverse solution satisfying all of the continuum mechanics requirements. As a result, less number of experimental measurements is required to reconstruct the complete residual fields. Consequently, the distribution of residual stresses is obtained without requiring the details of the hardening behavior of the material. Furthermore, the eigenstrain field is inversely determined satisfying the total strain compatibility equations, and a closed form analytical solution is presented for the distribution of eigenstrains.

Smooth trajectory planning for a parallel manipulator with joint friction and jerk constraints

In order to achieve better tracking accuracy effectively, a new smooth and near time-optimal trajectory planning approach is proposed for a parallel manipulator subject to kinematic and dynamic constraints. The complete dynamic model is constructed with consideration of all joint frictions. The presented planning problem can be solved efficiently by formulating a new limitation curve for dynamic constraints and a reduced form for jerk constraints. The motion trajectory is planned with quartic and quintic polynomial splines in Cartesian space and septuple polynomial splines in joint space. Experimental results show that smaller tracking error can be obtained. The developed method can be applied to any robots with analytical inverse kinematic and dynamic solutions.

Exact analytical inverse mapping of decomposable TS fuzzy systems with singleton and linear consequents

In this study, a new method is proposed for the exact analytical inverse mapping of Takagi–Sugeno fuzzy systems with singleton and linear consequents where the input variables are described by using strong triangular partitions. These fuzzy systems can be decomposed into several fuzzy subsystems. The output of the fuzzy subsystem results in multi-linear form in singleton consequent case or multi-variate second order polynomial form in linear consequent case. Since there exist explicit analytical formulas for the solutions of first and second order equations, the exact analytical inverse solutions can be obtained for decomposable Takagi–Sugeno fuzzy systems with singleton and linear consequents. In the proposed method, the output of the fuzzy subsystem is represented by using the matrix multiplication form. The parametric inverse definition of the fuzzy subsystem is obtained by using appropriate matrix partitioning with respect to the inversion variable. The inverse mapping of each fuzzy subsystem can then easily be calculated by substituting appropriate parameters of the fuzzy subsystem into this parametric inverse definition. So, it becomes very easy to find the analytical inverse mapping of the overall Takagi–Sugeno fuzzy system by composing inverse mappings of all fuzzy subsystems. The exactness and the effectiveness of the proposed inversion method are demonstrated on trajectory tracking problems by simulations.