Singularity Robust Inverse Research Articles

Singularity-robust full-pose workspace control of space manipulators with non-zero momentum

We address the end-effector full-pose tracking control problem in free-floating space manipulators, experiencing constant non-zero linear and angular momentum. The aim is to develop an output-tracking (workspace) control law free of singularities due to parameterizing the end-effector motion and being robust against singularities of the input–output decoupling matrix (generalized Jacobian matrix). Space manipulators are modeled as open-chain multi-body systems with single- and multi-degree-of-freedom joints, whose kinematics and dynamics are formulated on the Special Euclidean group SE(3). Such systems exhibit conserved (not necessarily zero) total momentum when operating in the free-floating regime, which we use to systematically reduce their dynamical equations by eliminating the base spacecraft’s motion. To avoid parameterizing the end-effector motion, we consider its full pose as the system output and develop a novel feedback linearization technique on the matrix Lie group SE(3) in the reduced phase space of the space manipulator. We then propose an intrinsic feedforward, feedback proportional–integral–derivative workspace controller involving a coordinate-free pose error function on SE(3) and velocity error on its Lie algebra. Using a Lyapunov candidate, this controller is proven to stabilize the end-effector pose to a feasible desired trajectory. The input–output decoupling matrix in the proposed control law can lose rank at some regions of the configuration space; hence, we implement a singularity-robust inverse, derived from the damped least squares method, to avoid impractical joint torques in these regions. The developed controller is implemented on a 7-degree-of-freedom manipulator onboard a spacecraft and its efficacy and robustness are demonstrated through a series of simulations.

Motion Planning for Vibration Reduction of a Railway Bridge Maintenance Robot with a Redundant Manipulator

Motivated by the potential applications of maintenance and inspection tasks for railway bridges, we have developed a biped climbing robot. The biped climbing robot can climb on the steel guardrail of the railway bridge with two electromagnetic feet and implement the maintenance and inspection tasks by a redundant manipulator with 7 degrees of freedom. To reduce the vibration of the manipulator caused by the low rigidity of the guardrail and the discontinuous trajectories of joints, a motion planning algorithm for vibration reduction is proposed in this paper. A geometric path accounting for obstacle avoidance and the manipulator’s center of gravity is determined by the gradient projection method with a singularity-robust inverse. Then, a piecewise quintic polynomial S shape curve with a smooth jerk (derivative of joint angular acceleration) profile is used to interpolate the sequence of joint angular position knots that are transformed from the via-points in the obstacle-avoidance path. The parameters of the quintic polynomial S-curve are determined by a nonlinear programming problem in which the objective function is to minimize the maximus of the torque exerted by the manipulator on the guardrail throughout the jerk-continuous trajectory. Finally, a series of simulation experiments are conducted to validate the effectiveness of the proposed algorithm. The simulation results show that the tracking errors of the trajectory with the proposed optimization algorithm are significantly smaller than the tracking errors of the trajectory without optimization. The absolute values of mean deviation of the tracking errors of the three coordinate axes decreased by at least 48.3% compared to the trajectory without vibration-reduction in the triangle working path and linear working path trajectory following simulations. The analysis results prove that the proposed algorithm can effectively reduce the vibration of the end effector of the manipulator.

Robust adaptive multi-task tracking control of redundant manipulators with dynamic and kinematic uncertainties and unknown disturbances

In this paper, a novel adaptive multi-priority controller for redundant manipulators is proposed to accomplish the multi-task tracking when kinematic/dynamic uncertainties and unknown disturbances exist. Prioritized redundancy resolution in kinematic level is incorporated into this passivity-based control framework. The kinematic and dynamic parameter adaptations are driven by both tracking error and prediction error. Moreover, the tracking information from both primary and subtasks are all utilized to accelerate the parameter estimation when the tasks are independent, whereas the inevitable tracking error of the subtasks due to algorithmic singularities is properly eliminated in the adaptation laws when the tasks are dependent. Potential ill-conditioned solution of the pseudoinverse is avoided using an improved singularity-robust inverse of the projected Jacobian. Along with the improvement of the multi-task tracking performance, smoothness of the commanded torques is still guaranteed for easy application. Measurements of the noisy joint acceleration and task velocity are avoided. The controller is mathematically derived based on Lyapunov stability analysis. Simulation results of the two cases are presented to verify the effectiveness and superiority of the proposed controller.

Evaluation of Steering Algorithm Optimality for Single-Gimbal Control Moment Gyroscopes

Analytic optimization methods typically used to derive optimal steering algorithms for single-gimbal control moment gyros do not consider the structure of the Jacobian matrix mapping the gimbal rates onto the desired torque within their cost function. Many of the steering algorithms resulting from these optimization methods systematically take first and second derivatives, forming the Jacobian and Hessian matrices to obtain a solution. However, the optimality is usually a local result and cannot be mapped back to its resulting performance. It is shown that the majority of steering algorithms are optimal with respect to one specific cost function previously published and that the design of the weighting matrices within the cost is what distinguishes steering algorithms. The author analytically shows how the blended inverse, Moore-Penrose pseudoinverse, generalized inverse steering law, singularity robust inverse, generalized singularity robust inverse, singular direction avoidance, local-gradient methods, and the hybrid steering logic are derived from the same optimizations but their sense of optimality is lost because the structure of singularities is not considered in the optimization process. In addition, the author also points out that the design of the quadratic costs weighting matrix used for optimization and desired gimbal rate is of the highest importance in differentiation between steering law performance.

Singularity Robust Steering Logic for Redundant Single-Gimbal Control Moment Gyros

In this paper we investigate a problem inherent in a system of redundant single-gimbal control moment gyros (CMGs). In particular, we present a simple yet effective way of passing through, and also escaping from, any internal singularities. The proposed approach is based on the so-called singularity robust inverse, but it utilizes a nondiagonal weighting matrix in the mixed, two-norm and weighted least-squares minimization problem. The proposed steering logic effectively generates deterministic dither signals when the CMG system becomes near singular. Because the proposed CMG steering logic is based on the simple minimum two-norm, pseudoinverse solution, it does not explicitly avoid encounters but it rather approaches and rapidly transits unavoidable singularities whenever needed. However, the proposed steering logic is not intended for special missions in which prescribed attitude trajectories are to be exactly tracked in the presence of internal singularities.

Singular inverse kinematic problem for robotic manipulators: a normal form approach

In this article, a new method of solving the singular inverse kinematic problem for nonredundant robotic manipulators is proposed, based on normal forms of the manipulator kinematics. A complete solution to the inverse problem is presented under assumption that the kinematics are equivalent to the quadratic normal form. Comparisons with other existing methods, including the singularity-robust inverse, are made that reveal advantages of the normal form approach. Experimental results concerned with singular tracking in the AdeptOne manipulator are reported.

Global steering of single gimballed control moment gyroscopes using a directed search

A guided depth-first search that manages null motion about torque-producing trajectories calculated with a singularity-robust inverse is proposed as a practical feedforward steering law that can globally avoid (or minimize the impact of) singular states in minimally redundant systems of single gimballed control moment gyroscopes. Cost and heuristic functions are defined to guide the search procedure in improving gimbal trajectories. On- orbit implementation of the steering law is proposed as an extension to momentum management algorithms. A set of simulation examples is presented, illustrating the search performance for a minimally redundant, pyramid- mounted array. Sensitivities o£ feedforward gimbal trajectories are examined in the presence of unmodeled disturbances, and techniques are proposed for avoiding excessive divergence. HE next generations of manned and unmanned space- craft will require enhanced control algorithms in order to efficiently achieve their proposed mission objectives. Reg- ularly coping with uncertainties in the orbital environment, dynamically changing spacecraft configurations (i.e., via docking and buildup), nonlinear actuator properties, and the need to tolerate potential hardware failures will mandate develop- ment of control strategies considerably beyond the available state-of-the-art. Because of the priority placed on minimizing cost, weight, and consumable requirements, future spacecraft will not always be able to rely on highly complex, multiply- redundant actuator systems (as is often now the case), but must employ more flexible and intelligent schemes that effi- ciently exploit all available onboard control capability. This study has examined methods of applying heuristic search techniques to perform adaptable inverse kinematics and ac- tuator management for spacecraft attitude control. In partic- ular, single gimballed control moment gyros (SGCMGs) were chosen as torque actuators. Although many existing al- gorithms may suffice for steering double gimballed CMG (DGCMG) arrays, such as envisioned for the NASA Space Station, no powerful techniques have been developed to man- age minimally redundant arrays of SGCMGs. These devices, however, are ideally suited as momentum-exchange effectors for a wide class of future spacecraft because of their large torque output and momentum capacity. They offer significant cost, power, weight, and reliability advantages over DGCMGs and could provide attitude control for a wide variety of future spacecraft, as torque requirements in many proposed mod- erate-sized vehicles may surpass the capability of available reaction wheels. Because of the complicated nonlinear map- ping between the input (gimbal) space and output (momen- tum) space, however, effective steering laws that reliably avoid problematic singular states have not been developed for min- imally redundant SGCMG systems, discouraging their appli- cation in many situations. To fully exploit the capability of SGCMGs, intelligent steering laws must be developed that address the system nonlinearities and avoid singular states over global trajectories.

Steering law design for redundant single-gimbal control moment gyroscopes

Two steering laws are presented for single-gimbal control moment gyroscopes. An approach using the Moore-Penrose pseudoinverse with a nondirectional null-motion algorithm is shown by example to avoid internal singularities for unidirectional torque commands, for which existing algorithms fail. Because this is still a tangent-based approach, however, singularity avoidance cannot be guaranteed. The singularity robust inverse is introduced as an alternative to the pseudoinverse for computing torque-producing gimbal rates near singular states. This approach, coupled with the nondirectional null algorithm, is shown by example to provide better steering law performance by allowing torque errors to be produced in the vicinity of singular states.

Inverse Kinematic Solutions With Singularity Robustness for Robot Manipulator Control

The singularity problem is an inherent problem in controlling robot manipulators with articulated configuration. In this paper, we propose to determine the joint motion for the requested motion of the endeffector by evaluating the feasibility of the joint motion. The determined joint motion is called an inverse kinematic solution with singularity robustness, because it denotes feasible solution even at or in the neighborhood of singular points. The singularity robust inverse (SR-inverse) is introduced as an alternative to the pseudoinverse of the Jacobian matrix. The SR-inverse of the Jacobian matrix provides us with an approximating motion close to the desired Cartesian trajectory of the endeffector, even when the inverse kinematic solution by the inverse or the pseudoinverse of the Jacobian matrix is not feasible at or in the neighborhood of singular points. The properties of the SR-inverse are clarified by comparing it with the inverse and the pseudoinverse. The computational complexity of the SR-inverse is considered to discuss its implementability. Several simulation results are also shown to illustrate the singularity problem and the effectiveness of the inverse kinematic solution with singularity robustness.