Particle Swarm Optimization for high-DOF inverse kinematics

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.

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.

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.

Computing joint commands in real time by solving an inverse kinematic (IK) problem is an important manipulation task that is commonly encountered in agricultural operations such as fruit harvesting, disease detection and leaf sampling. To date, different numerical and analytical approaches have been proposed, and many of them are able to take constraints into account. In this study, a set of analytical algorithms is investigated to quickly solve the constrained five-degree-of-freedom inverse kinematic problem. The constraints due to actuator limitations and mechanical design of the mechanism are utilised in generating the solution sets for joint variables. Also a suboptimal solution for the inverse kinematic problem within this set is derived. The proposed methods are compared with a typical numerical solution based on non-linear constrained optimisation. It is shown that the proposed analytical and suboptimal semi-analytical solutions can be found in a much quicker fashion as compared to the non-linear constrained optimisation approach. The algorithms are validated by the handling mechanism in a ground robot platform specifically designed to operate in strawberry orchards.

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 concerned with finding robot control parameters to bring the robot into a desired position under the kinematics and joint limit constraints. We present a globally optimal solution to the IK problem for a general serial 7DOF manipulator with revolute joints and a polynomial objective function. We show that the kinematic constraints due to rotations can be all generated by the second-degree polynomials. This is an important result since it significantly simplifies the further step where we find the optimal solution by Lasserre relaxations of nonconvex polynomial systems. We demonstrate that the second relaxation is sufficient to solve a general 7DOF IK problem. Our approach is certifiably globally optimal. We demonstrate the method on the 7DOF KUKA LBR IIWA manipulator and show that we are, in practice, able to compute the optimal IK or certify infeasibility in 99.9% tested poses. We also demonstrate that by the same approach, we are able to solve the IK problem for any generic (random) manipulator with seven revolute joints.

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.

In this study, inverse kinematics is solved for the European Robotic Arm. This robot is a type of space manipulator and has seven degrees of freedom. Forward and inverse kinematic is obtained by a new geometrical method. The limitations of robot workspace are calculated for robot joints and grippers. A geometrical inverse kinematic is presented for the first time, and the transferring process for various situations is developed to grasp the object at any position and orientation. Considering multiple missions, for robots in the transferring process, there isn’t a unit method to support all the situations and tasks in the inverse kinematic problem. To determine the workspace of robot, its geometry and object situation (orientation and position) are considered. To this end, using the suggested inverse kinematic algorithm, the target coordinate and orientations are obtained. In the presented inverse kinematic algorithm, an analytical method is used to derive the joint space variables in terms of workspace variables. The European Robotic Arm can move step by step from one target point to the next one. So, a general transferring algorithm is presented to realize the robot’s mission. In the presented algorithm, when the target is unreachable by the one-step operation, the transferring mission is utilized by the robot. Some simulation plans, to validate the proposed algorithms, indicate that the presented method works correctly.

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.

Robotic manipulation in surgical applications often demands the surgical instrument to pivot around a fixed point, known as remote center of motion (RCM). The RCM constraint ensures that the pivot point of the surgical tool remains stationary at the incision port, preventing tissue damage and bleeding. Precisely and efficiently controlling tool positioning and orientation under this constraint poses a complex Inverse Kinematics (IK) problem that must be solved in real-time to ensure patient safety and minimize complications. To address this problem, we propose PivotIK, an efficient IK solver that combines efficient evolutionary exploration with multi-objective Jacobian-based optimization. PivotIK can track desired tool poses accurately while satisfying the RCM constraint in real-time. We evaluated PivotIK through simulations and real-world experiments using redundant robotic manipulators with multi-degree-of-freedom surgical instruments. We compare PivotIK with other IK solvers in terms of solve rates, computation times, tracking errors, and RCM errors under various scenarios, including unconstrained and RCM-constrained trajectories. Our results show that PivotIK achieves superior performance, solving the IK problem in less than 1 ms with errors below 0.01 and 0.1 mm for tracking and RCM, respectively. Our real-world experiments confirm the effectiveness of PivotIK in ensuring smooth trajectory tracking and RCM compliance. PivotIK offers a promising solution for real-time IK for robotic manipulation under RCM constraints in surgical applications.

In this research, Particle Swarm Optimization (PSO) algorithm has been applied to solve the Inverse Kinematic (IK) problem for redundant serial manipulator robots. The new point of the study is to consider the continuity of the joints' variable. By noting this problem, the process of solving the Inverse Kinematic problem for the robots by the optimal algorithm has been significantly improved in terms of accuracy, execution time, and standard deviation (STD) as well as the number of iterations needed. The algorithm has been applied to solve the Inverse Kinematic problem on the 5-Degree of Freedom 5R robot model. The results show that just using the PSO algorithm with the above search domain limit has improved the quality of IK process.

In this study a five Degrees of Freedom (DOF) serial robot manipulator was designed and implemented. The inverse kinematics problem, which has not exact analytical solution (only one inverse kinematic solution for a predefined end effector position in three dimensional space), of the robot mechanism was solved by using the combination of the analytical method and a simple search method. In order to use the proposed method in real-time applications, the method is designed so that it can be used to solve the inverse kinematics problem for the next point of the end-effector while the mechanism is working. Moreover, so that to control the implemented mechanism, a user interface program was written by using Visual Basic programming language. Finally, the proposed inverse kinematic solution method was tested on two different trajectories, an arc shaped trajectory that composed of 58 points and a linear trajectory divided into 29 points. The obtained results revealed that the proposed method can be used successfully in solving the inverse kinematic problem of the designed mechanism.

Inverse kinematics is one of the most important and complicated problems in robotics, and there is almost no exact analytical solution for this problem. Alternatively, with significant growth in machine learning techniques in recent decades, numerical methods are widely being used to solve this problem. This article aims to present a novel application of two powerful meta-heuristic optimization algorithms including cuckoo optimization algorithm and imperialist competitive algorithm to solve robotic manipulators’ inverse kinematics problem for the first time. Recently, these two algorithms have been used to solve several problems in different majors more efficiently in comparison with other well-known algorithms. To validate the efficiency of proposed approaches and suggest them as the preferred numerical methods to solve this problem, a comprehensive study has been done on performance of almost all recently used numerical methods to solve the same problem including genetic algorithm, particle swarm optimization, harmony search and differential evolution algorithms as well as adaptive neuro-fuzzy inference system and two artificial neural networks (multilayer perceptron and radial basis function). Simulations have been performed on an anthropomorphic arm with spherical wrist manipulator case study because of its complex model and degree of freedom. Proposed approach can be included in computer-aided manufacturing packages to increase robotic manipulators’ efficiencies.

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.

Before moving a robot arm, it is of considerable interest to know whether there are any obstacles present in its path. Computer-based robots are usually served in the joint space, whereas objects to be manipulated are usually expressed in the Cartesian space because it is easier to visualize the correct end-effector position in Cartesian coordinates than in joint coordinates. In order to control the position of the end-effector of the robot, an inverse kinematics (IK) solution routine should be called upon to make the necessary conversion (Fu et al., 1987). Solving the inverse kinematics problem for serial robot manipulators is a difficult task; the complexity in the solution arises from the nonlinear equations (trigonometric equations) occurring during transformation between joint and Cartesian spaces. The common approach for solving the IK problem is to obtain an analytical close-form solution to the inverse transformation, unfortunately, close-form solution can only be found for robots of simple kinematics structure. For robots whose kinematics structures are not solvable in close-form; several numerical techniques have been proposed; however, there still remains several problems in those techniques such as incorrect initial estimation, convergence to the correct solution can not be guarantied, multiple solutions may exist and no solution could be found if the Jacobian matrix was in singular configuration (Kuroe et al., 1994; Bingual et al., 2005). A velocity singular configuration is a configuration in which a robot manipulator has lost at least one motion degree of freedom DOF. In such configuration, the inverse Jacobian will not exist, and the joint velocities of the manipulator will become unacceptably large that often exceed the physical limits of the joint actuators. Therefore, to analyze the singular conditions of a manipulator and develop effective algorithm to resolve the inverse kinematics problem in the singular configurations is of great importance (Hu et al., 2002). Many research efforts have been devoted towards solving this problem, one of the first algorithms employed was the Resolved Motion Rate-Control method (Whitney, 1969), which uses the pseudoinverse of the Jacobian matrix to obtain the joint velocities corresponding to a given end-effector velocity, an important drawback of this method was