Optimization Approach for Inverse Kinematic Solution

Inverse kinematics of serial or parallel manipulators can be computed from given Cartesian position and orientation of end effector and reverse of this would yield forward kinematics. Which is nothing but finding out end effector coordinates and angles from given joint angles. Forward kinematics of serial manipulators gives exact solution while inverse kinematics yields number of solutions. The complexity of inverse kinematic solution arises with the increment of degrees of freedom. Therefore it would be desired to adopt optimization techniques. Although the optimization techniques gives number of solution for inverse kinematics problem but it converses the best solution for the minimum function value. The selection of suitable optimization method will provides the global optimization solution, therefore, in this paper proposes quaternion derivation for 5R manipulator inverse kinematic solution which is later compared with teachers learner based optimization (TLBO) and genetic algorithm (GA) for the optimum convergence rate of inverse kinematic solution. An investigation has been made on the accuracies of adopted techniques and total computational time for inverse kinematic evaluations. It is found that TLBO is performing better as compared GA on the basis of fitness function and quaternion algebra gives better computational cost

Development of Alternative Methods for Robot Kinematics

The problem of finding mathematical tools to represent rigid body motions in space has long been on the agenda of physicists and mathematicians and is considered to be a well-researched and well-understood problem. Robotics, computer vision, graphics, and other engineering disciplines require concise and efficient means of representing and applying generalized coordinate transformations in three dimensions. Robotics requires systematic ways to represent the relative position or orientation of a manipulator rigid links and objects. However, with the advent of high-speed computers and their application to the generation of animated graphical images and control of robot manipulators, new interest arose in identifying compact and computationally efficient representations of spatial transformations. The traditional methods for representing forward kinematics of manipulators have been the homogeneous matrix in line with the D-H algorithm. In robotics, this matrix is used to describe one coordinate system with respect to another one. However for online operation and manipulation of the robotic manipulator in a flexible manner the computational time plays an important role. Although this method is used extensively in kinematic analysis but it is relatively neglected in practical robotic systems due to some complications in dealing with the problem of orientation representation. On the other hand, such matrices are highly redundant to represent six independent degrees of freedom. This redundancy can introduce numerical problems in calculations, wastes storage, and often increases the computational cost of algorithms. Keeping these drawbacks in mind, alternative methods are being sought by various researchers for representing the same and reducing the computational time to make the system fast responsive in a flexible environment. Researchers in robot kinematics tried alternative methods in order to represent rigid body transformations based on concepts introduced by mathematicians and physicists such as Euler angle or Epsilon algebra. In the present work alternative representations, using quaternion algebra and lie algebra are proposed, tried and compared

Geometric Algebra for Optimal Control with Applications in Manipulation Tasks

Many problems in robotics are fundamentally problems of geometry, which lead to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie algebra and dual quaternions. A unification and generalization of these popular formalisms can be found in geometric algebra. The aim of this paper is to showcase the capabilities of geometric algebra when applied to robot manipulation tasks. In particular the modelling of cost functions for optimal control can be done uniformly across different geometric primitives leading to a low symbolic complexity of the resulting expressions and a geometric intuitiveness. We demonstrate the usefulness, simplicity and computational efficiency of geometric algebra in several experiments using a Franka Emika robot. The presented algorithms were implemented in c++20 and resulted in the publicly available library \textit{gafro}. The benchmark shows faster computation of the kinematics than state-of-the-art robotics libraries.Comment: 16 pages, 13 figures

Computer algebra for solving dynamics problems of piezoelectric robots with large number of joints

The application of general control theory to complex mechanical systems represents an extremely difficult problem. If industrial piezoelectric robots have large number of joints, development of new control algorithms is unavoidable in order to achieve high positioning accuracy. The efficiency of computer algebra application was compared with the most popular methods of forming the dynamic equations of robots in real time. To this end, a computer algebra system VIBRAN was used. Expressions for the generalized inertia matrix of the robots have been derived by means of the computer algebra technique with the following automatic program code generation. As shown in the paper, such application could drastically reduce the number of floating point product operations that are required for efficient numerical simulation of piezoelectric robots

A review of parallel processing approaches to robot kinematics and Jacobian

Due to continuously increasing demands in the area of advanced robot control, it became necessary to speed up the computation. One way to reduce the computation time is to distribute the computation onto several processing units. In this survey we present different approaches to parallel computation of robot kinematics and Jacobian. Thereby, we discuss both the forward and the reverse problem. We introduce a classification scheme and classify the references by this scheme

Efficient Geometric Linearization of Moving-Base Rigid Robot Dynamics

The linearization of the equations of motion of a robotics system about a given state-input trajectory, including a controlled equilibrium state, is a valuable tool for model-based planning, closed-loop control, gain tuning, and state estimation. Contrary to the case of fixed based manipulators with prismatic or rotary joints, the state space of moving-base robotic systems such as humanoids, quadruped robots, or aerial manipulators cannot be globally parametrized by a finite number of independent coordinates. This impossibility is a direct consequence of the fact that the state of these systems includes the system's global orientation, formally described as an element of the special orthogonal group SO(3). As a consequence, obtaining the linearization of the equations of motion for these systems is typically resolved, from a practical perspective, by locally parameterizing the system's attitude by means of, e.g., Euler or Cardan angles. This has the drawback, however, of introducing artificial parameterization singularities and extra derivative computations. In this contribution, we show that it is actually possible to define a notion of linearization that does not require the use of a local parameterization for the system's orientation, obtaining a mathematically elegant, recursive, and singularity-free linearization for moving-based robot systems. Recursiveness, in particular, is obtained by proposing a nontrivial modification of existing recursive algorithms to allow for computations of the geometric derivatives of the inverse dynamics and the inverse of the mass matrix of the robotic system. The correctness of the proposed algorithm is validated by means of a numerical comparison with the result obtained via geometric finite difference

An Efficient Multi-solution Solver for the Inverse Kinematics of 3-Section Constant-Curvature Robots

Piecewise constant curvature is a popular kinematics framework for continuum robots. Computing the model parameters from the desired end pose, known as the inverse kinematics problem, is fundamental in manipulation, tracking and planning tasks. In this paper, we propose an efficient multi-solution solver to address the inverse kinematics problem of 3-section constant-curvature robots by bridging both the theoretical reduction and numerical correction. We derive analytical conditions to simplify the original problem into a one-dimensional problem. Further, the equivalence of the two problems is formalised. In addition, we introduce an approximation with bounded error so that the one dimension becomes traversable while the remaining parameters analytically solvable. With the theoretical results, the global search and numerical correction are employed to implement the solver. The experiments validate the better efficiency and higher success rate of our solver than the numerical methods when one solution is required, and demonstrate the ability of obtaining multiple solutions with optimal path planning in a space with obstacles.Comment: Robotics: Science and Systems 202