Neural Networks’ Based Inverse Kinematics Solution for Serial Robot Manipulators Passing Through Singularities

ABSTRACTSerial robotic manipulators find applications across diverse sectors such as industries, military, medical, space exploration, and underwater operations, where precision, efficiency, and safety are paramount. However, achieving effective control performance necessitates the effective resolution of the inverse kinematics (IK) problem inherent to manipulators. This challenge involves determining the requisite joint configurations to attain a desired endpoint position and orientation within the manipulator's workspace. The complexity arises due to the nonlinear equations and geometric relationships between cartesian space and joint space. Although various methods have been employed to address the IK problem in serial manipulators, there is a noticeable scarcity of publications comprehensively reviewing these techniques. Therefore, this paper undertakes an extensive review, thoroughly examining diverse IK methods applied in serial manipulators. This investigation of IK compiles literature spanning the last decade (2014–2023). Articles were collected from several web‐based repositories such as WoS, IEEE Xplore, ACM Digital Library, Google Scholar, and ScienceDirect, and a preferred reporting items for systematic reviews and meta‐analyses (PRISMA) protocol was adopted to acquire appropriate literature. The protocol involves examining the literature methodologies, major findings, and qualitative analysis. Thus, nearly 321 articles were screened and utilized in this review. Further, the IK methods are segregated into model‐driven methods (analytical, geometric, algebraic, numerical, and other model‐driven methods) and model‐free/data‐driven methods (metaheuristic (particle swarm optimization, genetic algorithm, and other metaheuristic methods), machine learning (artificial neural network, neuro‐fuzzy, and reinforcement learning), and other data‐driven methods). This paper offers a detailed exploration of limitations, challenges, performance comparisons, and future research directions of IK techniques. By conducting this study, we aim to assist prospective stakeholders in expanding their understanding of the remarkable potential of IK methods for serial manipulators.

The inverse kinematics (IK) problem is a fundamental problem in robotic manipulation. Traditional, Jacobian-based solutions to this problem are known to scale poorly with the number of degrees of freedom (DOF) in the manipulator, necessitating novel IK solutions for high-DOF manipulators. Metaheuristic optimization algorithms such as Particle Swarm Optimization (PSO) are a promising alternative approach to traditional IK techniques due to their strong performance on difficult and high-DOF problems in many diverse domains. Previous applications of PSO to the IK problem have focused on specific classes (e.g., planar) or models of manipulators or specific IK subproblems (e.g., position only IK). Furthermore, the experimental validation of these techniques has considered only manipulators with seven or fewer degrees of freedom and taken place almost exclusively in simulation. In this paper, we (1) generalize previous work to derive a fitness function that can be minimized to solve the full position and orientation IK problem for any serial manipulator while respecting joint limits and avoiding self-collisions, (2) present the first statistical analysis of PSO as a high-DOF IK solver on simulated manipulators with up to 180 DOF using this fitness function, and (3) present an important validation of PSO-based IK using this fitness function on real-world hardware on a difficult precision manipulation task.

This paper considers development of positioning systems for manipulator links to solve the forward kinematics problem (FKP) and inverse kinematics problem (IKP). Here we study a robotic manipulator with four degrees of freedom. It should be noted, that one of the relevant research problems of modern modular robotic devices consists in the lack of the universal algorithms, that would ensure kinematics problem recalculations in the cases of reconfigurations of the whole system. Challenges, the researchers are facing with when solving this problem, have to do with geometrical and non-linear equations (trigonometric equations), finding of inverse matrix of the Denavit—Hartenberg presentation, as well with other problems, such as multiple solutions when using the analytical approach. Common mathematical solutions of the inverse kinematics problem, such as geometric, iterative and algebraic ones, may not always lead to physically appropriate solutions. It’s also noteworthy, that, trying to introduce physical solutions for the manipulator, we need to take into account, that the number of calculation formulas increases, what, in turn, causes further computing power consumption increase. If the manipulator acquires additional degrees of freedom, analytical modeling becomes virtually impossible. One of relevant inverse kinematics solution methods consists in implementation of neural networks to that end. To solve this problem various sources were analyzed, considering alternative ways of target point discovery. Considering the analyzed papers, we propose to use a perceptron. Before training the network, we compose an algorithm, calculating the Denavit—Hartman presentation matrix and check for correctness of target point reach by the terminal manipulator link. We did calculations for a thousand positions of manipulator and object in the environment, fed to the neural network. When solving FKP we obtain object coordinates as network output, whereas in the case of IKP — manipulator link angles. We present kinematic scheme testing results, as well a control scheme for a manipulator with four degrees of freedom.

This paper presents a comparative study of analytical and deep learning (DL) approaches for solving the inverse kinematics (IK) problem in high-degree-of-freedom (DOF) robotic manipulators, focusing on the Lunar Exploration Rover System (LERS). The IK problem, central to robotic motion control, is traditionally solved using analytical methods, but recent advances in deep learning provide new opportunities to enhance precision and efficiency. The study evaluates the performance of DL Neural Networks, in addressing the complexities of high-DOF manipulators. A detailed comparison is conducted between traditional geometric solutions and DL-based models, with an emphasis on robustness and computational efficiency under noisy conditions. The results demonstrate the potential of DL methods to outperform traditional techniques in high-DOF environments, paving the way for future advancements in autonomous robotic systems. In addition to the IK study, the paper discusses the design and integration of the LERS robotic system, which plays a critical role in advancing autonomous lunar exploration under the ARTEMIS program. As part of this international collaboration involving NASA, the Canadian Space Agency (CSA), the European Space Agency (ESA), and the Japan Aerospace Exploration Agency (JAXA), LERS is tasked with supporting both crewed and unmanned missions on the Moon. LERS is designed to perform precise robotic manipulation tasks such as deploying infrastructure, gathering scientific data, and managing lunar resources, all of which are vital for future missions to Mars. The implementation of advanced IK solutions is key to enabling the precise control of LERS’s robotic arms, allowing for the successful execution of complex tasks such as assembling habitats, handling materials, and conducting scientific analyses on the lunar surface. This work highlights the importance of IK in ensuring that robotic systems like LERS can operate with the precision needed for the next generation of lunar missions.

Six degrees-of-freedom (DoF) robot manipulators are dynamically coupled and highly nonlinear multi-variable systems. Calculating the optimal joint angles from the operational space (i.e. inverse kinematics) and designing the optimal joint controller parameters are two important research topics. This paper proposes a 3-stage Particle Swarm Optimization (PSO)-based methodology for solving the inverse kinematics and optimizing the controller parameters. In the first stage, a PSO algorithm solves the inverse kinematics problem by minimizing a multi-objective cost function in the operational space (i.e. the error in the end-effector’s pose) and therefore finds the optimal joint angles. In the second stage, polynomial functions generate the desired trajectory between the initial and final poses in the joint space. Finally, a second PSO algorithm tunes six proportional-derivative (PD) controllers, one for each joint, to track the desired trajectory by minimizing another multi-objective cost function in the joint space. Two case studies, based on six DoF Puma 560, validate the performance of the proposed methodology. Simulation results show that the proposed 3-stage methodology provides fast and accurate results as the PSO algorithms are effective in solving the inverse kinematics problem and tuning the optimal PD parameters.

The aim of this paper is to characterize the notion of aspect in the workspace and in the joint space for parallel manipulators. In opposite to the serial manipulators, the parallel manipulators can admit not only multiple inverse kinematic solutions, but also multiple direct kinematic solutions. The notion of aspect introduced for serial manipulators in [Borrel 86], and redefined for parallel manipulators with only one inverse kinematic solution in [Wenger 1997], is redefined for general fully parallel manipulators. Two Jacobian matrices appear in the kinematic relations between the joint-rate and the Cartesian-velocity vectors, which are called the "inverse kinematics" and the "direct kinematics" matrices. The study of these matrices allow to respectively define the parallel and the serial singularities. The notion of working modes is introduced to separate inverse kinematic solutions. Thus, we can find out domains of the workspace and the joint space exempt of singularity. Application of this study is the moveability analysis in the workspace of the manipulator as well as path-planing and control. This study is illustrated in this paper with a RR-RRR planar parallel manipulator.

Robotic manipulators inevitably encounter singular configurations in the process of movement, which seriously affects their performance. Therefore, the identification of singular configurations is extremely important. However, serial manipulators that do not meet the Pieper criterion cannot obtain singular configurations through analytical methods. A joint angle parameterization method, used to obtain singular configurations, is here creatively proposed. First, an analytical method based on the Jacobian determinant and the proposed method were utilized to obtain their respective singular configurations of the Stanford manipulator. The singular configurations obtained through the two methods were consistent, which suggests that the proposed method can obtain singular configurations correctly. Then, the proposed method was applied to a seven-degree-of-freedom (7-DOF) serial manipulator and a planar 5R parallel manipulator. Finally, the correctness of the singular configurations of the 7-DOF serial manipulator was verified through the shape of the end-effector velocity ellipsoid, the value of the determinant, the value of the condition number, and the value of the manipulability measure. The correctness of singular configurations of the planar 5R parallel manipulator was verified through the value of the determinant, the value of the condition number, and the value of the manipulability measure.

The inverse kinematics determines the joint angles that result in the desired position of the manipulator's end-effector with regard to the reference coordinated system. The inverse kinematics solution is difficult a time that the mapping between the Cartesian space and the joint space is nonlinear and involves equations that can have multiple solutions. This work presents the application of a neural network with radial basis function (RBF) for the inverse kinematics solution of a robotic manipulator with three degrees of freedom. For a good learning generalization in the training phase of RBF network, some initial and final points had been initially generated inside of the manipulator's work volume. In order to prevent extreme angular oscillations, the angles for each one of the n generated points, they had been optimized by genetic algorithms. In accordance with the cartesian coordinate (x, y) supplied, the RBF neural network implemented in this work supplied angles that it had presented very next to the desired values.

Imitation learning is a way to teach robots skills that are demonstrated by humans. Transfering skills between these different kinematic structures seems to be straightforward in Cartesian space. Because of the correspondence problem, however, the result will most likely not be identical. This is why refinement is required, for example, by policy search. Policy search in Cartesian space is prone to reachability problems when using conventional inverse kinematic solvers. We propose a configurable approximate inverse kinematic solver and show that it can accelerate the refinement process considerably. We also compare empirically refinement in Cartesian space and refinement in joint space.

Minimally invasive surgery has undergone significant advancements in recent years, transforming various surgical procedures by minimizing patient trauma, postoperative pain, and recovery time. However, the use of robotic systems in minimally invasive surgery introduces significant challenges related to the control of the robot's motion and the accuracy of its movements. In particular, the inverse kinematics (IK) problem is critical for robot-assisted minimally invasive surgery (RMIS), where satisfying the remote center of motion (RCM) constraint is essential to prevent tissue damage at the incision point. Several IK strategies have been proposed for RMIS, including classical inverse Jacobian IK and optimization-based approaches. However, these methods have limitations and perform differently depending on the kinematic configuration. To address these challenges, we propose a novel concurrent IK framework that combines the strengths of both approaches and explicitly incorporates RCM constraints and joint limits into the optimization process. In this paper, we present the design and implementation of concurrent inverse kinematics solvers, as well as experimental validation in both simulation and real-world scenarios. Concurrent IK solvers outperform single-method solvers, achieving a 100% solve rate and reducing the IK solving time by up to 85% for an endoscope positioning task and 37% for a tool pose control task. In particular, the combination of an iterative inverse Jacobian method with a hierarchical quadratic programming method showed the highest average solve rate and lowest computation time in real-world experiments. Our results demonstrate that concurrent IK solving provides a novel and effective solution to the constrained IK problem in RMIS applications.

In this paper, we present a tutorial report of the literature on the damped-least squares method which has been used for computing velocity inverse kinematics of robotic manipulators. This is a local optimization method that can prevent infeasible joint velocities near singular configurations by using a damping factor to control the norm of the joint velocity vector. However, the exactness of the inverse kinematic solution has to be sacrificed in order to achieve feasibility. The damping factor is an important parameter in this technique since it determines the trade-off between the accuracy and feasibility of the inverse kinematic solution. Various methods that have been proposed to compute an appropriate damping factor are described. Redundant manipulators, possessing extra degrees of freedom, afford more choice of inverse kinematic solutions than do non-redundant ones. The damped least-squares method has been used in conjunction with redundancy resolution schemes to compute feasible joint velocities for redundant arms while performing an additional subtask. We outline the different techniques that have been proposed to achieve this objective. In addition, we introduce an iterative method to compute the optimal damping factor for one of the redundancy resolution techniques.

PurposeThis paper aims to develop a new real‐time effective method for solving the inverse kinematics (IK) problem, especially for those manipulators with high‐dimensional nonlinear kinematic equations.Design/methodology/approachThe paper transforms the IKs problem into a minimization problem. Then, a novel meta‐heuristic algorithm, called the electromagnetism‐like method (EM), is used to solve this equivalent problem. Moreover, in order to further improve the computational efficiency and accuracy of EM, a hybrid method which combines EM with the Davidon‐Fletcher‐Powell (DFP) method is proposed.FindingsThe results showed that EM is a powerful yet easy algorithm for solving the IKs problem of robot manipulators. Its complexity is independent on the characteristics of the kinematic equations involving dimensionality and the degree of nonlinearity. Moreover, EM can be used as an accompanying algorithm for DFP method to get better precision at a lower iteration number.Originality/valueThe method developed in this paper is a generalized approach that is efficient enough to obtain IK solutions independent of robot geometry and the number of degrees of freedom.

The inverse kinematics problem is ill-posed near kinematic singularities leading to extremely large joint velocities. This classical problem is addressed from a different perspective; instead of prescribing the end-effector (EE) motion in terms of a path parameter, the motion is parameterized in terms of the arc length in joint space. This allows tackling the inverse kinematics problem when approaching kinematic singularities. Thereupon a singularity consistent sampling scheme can easily be devised. A sampling of the arc length gives direct control over the joint motion for prescribed EE motion. The arc length parameterization can be used for solving the (time) optimal path following problem, avoiding the ill-conditioning upfront.KeywordsSerial robotsInverse kinematicsSingularitiesPath planningArc length parameterizationNon-equidistant discretization

We propose a geometric method to solve inverse kinematics (IK) problems of 7-DoF manipulators with joint offsets at shoulder, elbow, and wrist. Traditionally, inverse position kinematics for redundant manipulators are solved by using an iterative method based on the pseudo-inverse of the manipulator Jacobian. This provides a single solution among the infinitely many possible solutions for the IK problem of redundant manipulators. There are no closed-form IK solutions for redundant manipulators with multiple joint offsets. Using our method we can compute multiple IK solutions using two-parameter search by exploiting geometry of the structure of a redundant manipulator. Our proposed IK algorithm can handle multiple joint offsets and is mathematically simple to implement in a few lines of code. We apply our algorithm to compute IK solutions for 7-DoF redundant Baxter robot (that has joint offsets at shoulder, wrist, and elbow joints) for end-effector configurations where existing geometry-based IK solvers fail to find solutions. We also demonstrate the use of our algorithm in an application where we want to compute an IK solution (among the infinitely many possible solutions) that has minimum error bound in end-effector position, in the presence of random joint actuation and sensing uncertainties.

When the robot interpolates the trajectory at high speed, the joint will exceed the constraint due to the constraints of driving mechanism performance and other factors. Based on the analysis of driver constraints, an adaptive trajectory planning method based on joint constraints is proposed. Firstly, maximum ranges of joint velocities in joint space are analyzed in advance by using the look-ahead strategy for speed control. Secondly, a maximum runnable resultant velocity in this path is calculated based on the linear relationship between joint velocity in joint space and path velocity in Cartesian space. Finally, the path velocity in Cartesian space is used for interpolation. The simulation and experimental results show that compared with the traditional method, the proposed Trajectory planning method can give full play to the driving performance of the robot joints while ensuring the real-time performance, taking into account the operating efficiency and constraint requirements of the robot.