Riemannian Optimization for Distance-Geometric Inverse Kinematics

Solving the inverse kinematics problem is a fundamental challenge in motion planning, control, and calibration for articulated robots. Kinematic models for these robots are typically parametrized by joint angles, generating a complicated mapping between the robot configuration and the end-effector pose. Alternatively, the kinematic model and task constraints can be represented using invariant distances between points attached to the robot. In this paper, we formalize the equivalence of distance-based inverse kinematics and the distance geometry problem for a large class of articulated robots and task constraints. Unlike previous approaches, we use the connection between distance geometry and low-rank matrix completion to find inverse kinematics solutions by completing a partial Euclidean distance matrix through local optimization. Furthermore, we parametrize the space of Euclidean distance matrices with the Riemannian manifold of fixed-rank Gram matrices, allowing us to leverage a variety of mature Riemannian optimization methods. Finally, we show that bound smoothing can be used to generate informed initializations without significant computational overhead, improving convergence. We demonstrate that our inverse kinematics solver achieves higher success rates than traditional techniques, and substantially outperforms them on problems that involve many workspace constraints.Comment: 20 pages, 14 figure

Inverse Kinematics Based on Fuzzy Logic and Neural Networks for the WAM-Titan II Teleoperation System

The inverse kinematic problem is crucial for robotics. In this paper, a solution algorithm is presented using artificial intelligence to improve the pseudo-inverse Jacobian calculation for the 7-DOF Whole Arm Manipulator (WAM) and 6-DOF Titan II teleoperation system. An investigation of the inverse kinematics based on fuzzy logic and artificial neural networks for the teleoperation system was undertaken. Various methods such as Adaptive Neural-Fuzzy Inference System (ANFIS), Genetic Algorithms (GA), Multilayer Perceptrons (MLP) Feedforward Networks, Radial Basis Function Networks (RBF) and Generalized Regression Neural Networks (GRNN) were tested and simulated using MATLAB. Each method for identification of the pseudo-inverse problem was tested, and the best method was selected from the simulation results and the error analysis. From the results, the Multilayer Perceptrons with Levenberg-Marquardt (MLP-LM) method had the smallest error and the fastest computation among the other methods. For the WAM-Titan II teleoperation system, the new inverse kinematics calculations for the Titan II were simulated and analyzed using MATLAB. Finally, extensive C code for the alternative algorithm was developed, and the inverse kinematics based on the artificial neural network with LM method is implemented in the real system. The maximum error of Cartesian position was 1.3 inches, and from several trajectories, 75 % of time implementation was achieved compared to the conventional method. Because fast performance of a real time system in the teleoperation is vital, these results show that the new inverse kinematics method based on the MLP-LM is very successful with the acceptable error

Algorithmic differentiation improves the computational efficiency of OpenSim-based trajectory optimization of human movement

Algorithmic differentiation (AD) is an alternative to finite differences (FD) for evaluating function derivatives. The primary aim of this study was to demonstrate the computational benefits of using AD instead of FD in OpenSim-based trajectory optimization of human movement. The secondary aim was to evaluate computational choices including different AD tools, different linear solvers, and the use of first- or second-order derivatives. First, we enabled the use of AD in OpenSim through a custom source code transformation tool and through the operator overloading tool ADOL-C. Second, we developed an interface between OpenSim and CasADi to solve trajectory optimization problems. Third, we evaluated computational choices through simulations of perturbed balance, two-dimensional predictive simulations of walking, and three-dimensional tracking simulations of walking. We performed all simulations using direct collocation and implicit differential equations. Using AD through our custom tool was between 1.8 ± 0.1 and 17.8 ± 4.9 times faster than using FD, and between 3.6 ± 0.3 and 12.3 ± 1.3 times faster than using AD through ADOL-C. The linear solver efficiency was problem-dependent and no solver was consistently more efficient. Using second-order derivatives was more efficient for balance simulations but less efficient for walking simulations. The walking simulations were physiologically realistic. These results highlight how the use of AD drastically decreases computational time of trajectory optimization problems as compared to more common FD. Overall, combining AD with direct collocation and implicit differential equations decreases the computational burden of trajectory optimization of human movement, which will facilitate their use for biomechanical applications requiring the use of detailed models of the musculoskeletal system.Postprint (published version

Multi-Contact Postures Computation on Manifolds

International audienceWe propose a framework to generate static robot configurations satisfying a set of physical and geometrical constraints. This is done by formulating nonlinear constrained optimization problems over non-Euclidean manifolds and solving them. To do so, we present a new sequential quadratic programming (SQP) solver working natively on general manifolds, and propose an interface to easily formulate the problems, with the tedious and error-prone work automated for the user. We also introduce several new types of constraints for having more complex contacts or working on forces/torques. Our approach allows an elegant mathematical description of the constraints and we exemplify it through formulation and computation examples in complex scenarios with humanoid robots

A second order minimum-energy filter on the special orthogonal group

Abstract— This work documents a case study in the application of Mortensen’s nonlinear filtering approach to invariant systems on general Lie groups. In this paper, we consider the special orthogonal group SO(3) of all rotation matrices. We identify the exact form of the kinematics of the minimumenergy (optimal) observer on SO(3) and note that it depends on the Hessian of the value function of the associated optimal control problem. We derive a second order approximation of the dynamics of the Hessian by neglecting third order terms in the expansion of the dynamics. This yields a Riccati equation that together with the optimal observer equation form a second order minimum-energy filter on SO(3). The proposed filter is compared to the multiplicative extended Kalman filter (MEKF), arguably the industry standard for attitude estimation, by means of simulations. Our studies indicate superior transient and asymptotic tracking performance of the proposed filter as compared to the MEKF