Discussions on Inverse Kinematics based on Levenberg-Marquardt Method and Model-Free Adaptive (Predictive) Control

In this brief, the current robust numerical solution to the inverse kinematics based on Levenberg-Marquardt (LM) method is reanalyzed through control theory instead of numerical method. Compared to current works, the robustness of computation and convergence performance of computational error are analyzed much more clearly by analyzing the control performance of the corrected model free adaptive control (MFAC). Then mainly motivated by minimizing the predictive tracking error, this study suggests a new method of model free adaptive predictive control (MFAPC) to solve the inverse kinematics problem. At last, we apply the MFAPC as a controller for the robotic kinematic control problem in simulation. It not only shows an excellent control performance but also efficiently acquires the solution to inverse kinematic

Mirror-Descent Inverse Kinematics for Box-constrained Joint Space

This paper proposes a new Jacobian-based inverse kinematics (IK) explicitly considering box-constrained joint space. To control humanoid robots, the reference pose of end effector(s) is planned in task space, then mapped into the reference joints by IK. Due to the limited analytical solutions for IK, iterative numerical IK solvers based on Jacobian between task and joint spaces have become popular. However, the conventional Jacobian-based IK does not explicitly consider the joint constraints, and therefore, they usually clamp the obtained joints during iteration according to the constraints in practice. The problem in clamping operation has been pointed out that it causes numerical instability due to non-smoothed objective function. To alleviate the clamping problem, this study explicitly considers the joint constraints, especially the box constraints in this paper, inside the new IK solver. Specifically, instead of clamping, a mirror descent (MD) method with box-constrained real joint space and no-constrained mirror space is integrated with the conventional Jacobian-based IK methods, so-called MD-IK. In addition, to escape local optima nearly on the boundaries of constraints, a heuristic technique, called $\epsilon$-clamping, is implemented as margin in software level. As a result, MD-IK achieved more stable and enough fast i) regulation on the random reference poses and ii) tracking to the random trajectories compared to the conventional IK solvers.Comment: 7 pages, 6 figure

Discussion on a Class of Model-Free Adaptive Control for Multivariable Systems

The model-free adaptive control (MFAC) law is a promising method in applications. We analyzed model-free adaptive control (MFAC) law through closed-loop function to widen its application range

Learning Constrained Distributions of Robot Configurations with Generative Adversarial Network

In high dimensional robotic system, the manifold of the valid configuration space often has a complex shape, especially under constraints such as end-effector orientation or static stability. We propose a generative adversarial network approach to learn the distribution of valid robot configurations under such constraints. It can generate configurations that are close to the constraint manifold. We present two applications of this method. First, by learning the conditional distribution with respect to the desired end-effector position, we can do fast inverse kinematics even for very high degrees of freedom (DoF) systems. Then, we use it to generate samples in sampling-based constrained motion planning algorithms to reduce the necessary projection steps, speeding up the computation. We validate the approach in simulation using the 7-DoF Panda manipulator and the 28-DoF humanoid robot Talos

Second-order and implicit methods in numerical integration improve tracking performance of the closed-loop inverse kinematics algorithm

A general approach to solve the inverse kinematics problem of series manipulators, i.e. finding the required joint motions for the desired end effector motions, is based on the linear approximation of the forward kinematics map and discretization of the continuous problem. Due to the linearization, first velocities are calculated, so numerical integration needs to be done to get the joint variables. This general solution is just a numerical approximation, thus improving the tracking performance of the inverse kinematics algorithm is of great importance. The application of several numerical integration techniques (implicit Euler, explicit trapezoid, implicit trapezoid) is analyzed, and a fix point iteration is given that can be used to calculate implicit solutions. The tracking performance of the spatial inverse positioning problem of a spatial manipulator is analyzed by checking the tracking error in the desired direction (i.e. along the derivative of the desired end effector path) and in the plane perpendicular to the desired direction. The application of the explicit and implicit trapezoid methods yielded much better tracking performance in the directions orthogonal to the desired direction when the end effector had to track a linear path, while the tracking performance in the desired direction was similar for all the methods. Simulations showed that the application of implicit and second-order methods in the numerical integration may greatly improve the tracking performance of the closed-loop inverse kinematics algorithm

Inverse kinematics of a humanoid robot with non-spherical hip: a hybrid algorithm approach

This paper describes an approach to solve the inverse kinematics problem of humanoid robots whose construction shows a small but non negligible offset at the hip which prevents any purely analytical solution to be developed. Knowing that a purely numerical solution is not feasible due to variable efficiency problems, the proposed one first neglects the offset presence in order to obtain an approximate “solution” by means of an analytical algorithm based on screw theory, and then uses it as the initial condition of a numerical refining procedure based on the Levenberg‐Marquardt algorithm. In this way, few iterations are needed for any specified attitude, making it possible to implement the algorithm for real‐time applications. As a way to show the algorithm’s implementation, one case of study is considered throughout the paper, represented by the SILO2 humanoid robot

Modeling robot geometries like molecules, application to fast multi-contact posture planning for humanoids

Traditional joint-space models used to describe equations of motion for humanoid robots offer nice properties linked directly to the way these robots are built. However, from a computational point of view and convergence properties, these models are not the fastest when used in planning optimizations. In this paper, inspired by Cartesian coordinates used to model molecular structures, we propose a new modeling technique for humanoid robots. We represent robot segments by vectors and derive equations of motion for the full body. Using this methodology in a complex task of multi-contact posture planning with minimal joint torques, we set up optimization problems and analyze the performance. We demonstrate that compared to joint-space models that get trapped in local minima, the proposed vector-based model offers much faster computational speed and a suboptimal but unique final solution. The underlying principle lies in reducing the nonlinearity and exploiting the sparsity in the problem structure. Apart from the specific case study of posture optimization, these principles can make the proposed technique a promising candidate for many other optimization-based complex tasks in robotics

Benchmarking Stability of Bipedal Locomotion Based on Individual Full Body Dynamics and Foot Placement Strategies–Application to Impaired and Unimpaired Walking

The principles underlying smooth and effortless human walking while maintaining stability as well as the ability to quickly respond to unexpected perturbations result from a plethora of well-balanced parameters, most of them yet to be determined. In this paper, we investigate criteria that may be useful for benchmarking stability properties of human walking. We perform dynamic reconstructions of human walking motions of unimpaired subjects and subjects walking with transfemoral prostheses from motion capture recordings using optimal control. We aim at revealing subject-specific strategies in applying dynamics in order to maintain steady gait considering irregularities such as deviating gait patterns or asymmetric body segment properties. We identify foot placement with respect to the Instantaneous Capture Point as the strategy globally applied by the subjects to obtain steady gait and propose the Residual Orbital Energy as a measure allowing for benchmarking human-like gait toward confident vs. cautious gait

Fast Nonlinear Least Squares Optimization of Large-Scale Semi-Sparse Problems

Many problems in computer graphics and vision can be formulated as a nonlinear least squares optimization problem, for which numerous off-the-shelf solvers are readily available. Depending on the structure of the problem, however, existing solvers may be more or less suitable, and in some cases the solution comes at the cost of lengthy convergence times. One such case is semi-sparse optimization problems, emerging for example in localized facial performance reconstruction, where the nonlinear least squares problem can be composed of hundreds of thousands of cost functions, each one involving many of the optimization parameters. While such problems can be solved with existing solvers, the computation time can severely hinder the applicability of these methods. We introduce a novel iterative solver for nonlinear least squares optimization of large-scale semi-sparse problems. We use the nonlinear Levenberg-Marquardt method to locally linearize the problem in parallel, based on its firstorder approximation. Then, we decompose the linear problem in small blocks, using the local Schur complement, leading to a more compact linear system without loss of information. The resulting system is dense but its size is small enough to be solved using a parallel direct method in a short amount of time. The main benefit we get by using such an approach is that the overall optimization process is entirely parallel and scalable, making it suitable to be mapped onto graphics hardware (GPU). By using our minimizer, results are obtained up to one order of magnitude faster than other existing solvers, without sacrificing the generality and the accuracy of the model. We provide a detailed analysis of our approach and validate our results with the application of performance-based facial capture using a recently-proposed anatomical local face deformation model

Dynamic whole-body motion generation under rigid contacts and other unilateral constraints

The most widely used technique for generating wholebody motions on a humanoid robot accounting for various tasks and constraints is inverse kinematics. Based on the task-function approach, this class of methods enables the coordination of robot movements to execute several tasks in parallel and account for the sensor feedback in real time, thanks to the low computation cost. To some extent, it also enables us to deal with some of the robot constraints (e.g., joint limits or visibility) and manage the quasi-static balance of the robot. In order to fully use the whole range of possible motions, this paper proposes extending the task-function approach to handle the full dynamics of the robot multibody along with any constraint written as equality or inequality of the state and control variables. The definition of multiple objectives is made possible by ordering them inside a strict hierarchy. Several models of contact with the environment can be implemented in the framework. We propose a reduced formulation of the multiple rigid planar contact that keeps a low computation cost. The efficiency of this approach is illustrated by presenting several multicontact dynamic motions in simulation and on the real HRP-2 robot