Null Space Solution Research Articles

An optical-scan method for measuring the as installed surface port incidence angles for flush air data sensing (FADS) systems

The Flush Air Data Sensing (FADS) System, where air data are inferred from non-intrusive surface pressure measurements, uses natural contours of the vehicle forebody, wing leading edge, or probe. Although multiple methods have been developed to derive airdata from the sensed pressure matrix, all methods rely on accurate knowledge of local surface contours at the port locations. One of the most well-developed solution methods curve-fits the surface pressure distribution against the associated surface incidence angles using a quasi-Newtonian model. The well-known "Triples" algorithm extracts airdata from the curve-fit model. This solution method requires precise knowledge of as-installed incidence angles, i.e. the angles between the surface normal and the longitudinal axis of the vehicle. This study investigates the feasibility and accuracy of using an inexpensive optical-scanning system to measure the in-situ FADS pressure ports surface incidence angles. Here, two legacy 3-D printed probe shapes, as previously tested during a series of very low-speed wind tunnel tests, were used to develop and evaluate this method. The shapes 1) a hemispherical head cylindrical forebody, and 2) a Rankine-Body, were scanned along the longitudinal axis and the resulting point-cloud was edited using open-source software to generate three concentric "loops" surrounding each surface port. Each annular loop was assumed as co-planar with the surface port, and the singular-value decomposition (SVD) was used calculate the local surface gradient vector from the null-space solution. From the resulting gradient vector, geometric relationships calculate the port's polar coordinates including the surface incidence angle. For both body contours the resulting calculations are compared to the "known" design surface angles as prescribed for the 3-D prints. Error plots are presented for each individual ring-set, and for the collected set using all three ring together. For the collected data sets, the incidence angle calculations are accurate to within a quarter-degree.

Adaptive Control Strategy of a 2-Axes Gimbal Structure Loading Device for a Stair-Climbing Delivery Robot

This paper studies an adaptive control strategy for a 2-axes gimbal structure loading device for a stair-climbing delivery robot. The properties of the loading device change depending on the luggage, and the loaded luggage cannot be specified. It is difficult to design an appropriate controller with fixed PID gains. To optimize the controller regardless of luggage condition, the adaptive control strategy is conducted by system identification and gain scheduling. The system identification estimates the parameters of the dynamic equations. The gain scheduler optimizes the PID controller using the estimated parameters. The system identification technique is based on the least-squares method. The accuracy of system identification is improved by a null-space solution. The gain scheduler consists of surface functions defined by interpolation of the optimized PID gains. The system identification techniques are verified by simulation, comparing a specific system assumed as a real system and the estimated system. The experiments using the motion platform verify the adaptive control strategy. This proposed control strategy adapts the controller to the system of the loading device.

First-Passage Problem in Random Vibrations With Radial Basis Function Neural Networks

Abstract The first-passage time probability plays an important role in the reliability assessment of dynamic systems in random vibrations. To find the solution of the first-passage time probability is a challenging task. The analytical solution to this problem is not available even for linear dynamic systems. For nonlinear and multi-degree-of-freedom systems, it is even more challenging. This paper proposes a radial basis function neural networks method for solving the first-passage time probability problem of linear, nonlinear, and multi-degree-of-freedom dynamic systems. In this paper, the proposed method is applied to solve for the backward Kolmogorov equation subject to boundary conditions defined by the safe domain. A null-space solution strategy is proposed to deal with the boundary condition. Several examples including a two degrees-of-freedom nonlinear Duffing system are studied with the proposed method. The results are compared with Monte Carlo simulations. It is believed that the radial basis function neural networks method provides a new and effective tool for the reliability assessment and design of multi-degree-of-freedom nonlinear stochastic dynamic systems.

Utilization of the Moore-Penrose inverse in the modeling of overconstrained mechanisms with frictionless and frictional joints

The indeterminate equations that describe overconstrained mechanisms are often solved using the Moore-Penrose inverse. Some limitations of this approach are investigated here. Firstly, frictionless systems are considered. The problem of solvability of accelerations and joint reactions is studied—the non-uniqueness of reactions and uniqueness of accelerations is discussed. Next, the dependence of the results on the selection of the physical units is examined. It is checked which elements of the solution are physically non-equivalent after changing the units; relationships between different solutions are derived. Secondly, frictional systems are considered. Joint friction dependent and independent on normal load is studied. Fixed point iterations and Newton's method are applied to solve nonlinear equations of motion. The Moore-Penrose inverse is employed to conduct calculations. The null space solution components are considered, and necessary amendments in the iterative processes termination criteria are discussed. The origins of non-uniqueness of accelerations and Lagrange multipliers are analyzed. The unit-sensitivity of frictional system models is addressed. Finally, an illustrative example is given, and conclusions are drawn—limitations and possible improvements of the Moore-Penrose inverse approach are discussed.

A Unified Approach to Exact and Approximate Motion Synthesis of Spherical Four-Bar Linkages Via Kinematic Mapping

This paper studies the problem of spherical four-bar motion synthesis from the viewpoint of acquiring circular geometric constraints from a set of prescribed spherical poses. The proposed approach extends our planar four-bar linkage synthesis work to spherical case. Using the image space representation of spherical poses, a quadratic equation with ten linear homogeneous coefficients, which corresponds to a constraint manifold in the image space, can be obtained to represent a spherical RR dyad. Therefore, our approach to synthesizing a spherical four-bar linkage decomposes into two steps. First, find a pencil of general manifolds that best fit the task image points in the least-squares sense, which can be solved using singular value decomposition (SVD), and the singular vectors associated with the smallest singular values are used to form the null-space solution of the pencil of general manifolds; second, additional constraint equations on the resulting solution space are imposed to identify the general manifolds that are qualified to become the constraint manifolds, which can represent the spherical circular constraints and thus their corresponding spherical dyads. After the inverse computation that converts the coefficients of the constraint manifolds to the design parameters of spherical RR dyad, spherical four-bar linkages that best navigate through the set of task poses can be constructed by the obtained dyads. The result is a fast and efficient algorithm that extracts the geometric constraints associated with a spherical motion task, and leads naturally to a unified treatment for both exact and approximate spherical motion synthesis.

Positioning control of an underwater robot with tilting thrusters via decomposition of thrust vector

Positioning control of an underwater robot is a challenging problem due to the high disturbances of ocean flow. To overcome the high disturbance, a new underwater robot with tilting thrusters was proposed previously, which can compensate for disturbance by focusing the thrusting force in the direction of the disturbance. However, the tilting motion of the thrusters makes the system nonlinear, and the limited tilting speed sometimes makes the robot unstable. Therefore, an optimized controller is necessary. A new positioning controller is proposed for this robot using a vector decomposition method. Based on the dynamic model, the nonlinear force input term of the tilting thrusters is decomposed in the horizontal and vertical directions. Based on the decomposition, the solution is determined by a pseudo-inverse and null-space solution. Using the characteristics of the decomposed input matrix, the final solution can be found by solving a simple second-order algebraic equation to overcome the limitations of the tilting speed. The positioning was simulated to validate the proposed controller by comparing the results with a switching-based controller. Tracking results are also presented. In future work, a high-level control strategy will be developed to take advantage of the tilting thrusters by focusing the forcing direction toward the disturbance with a limited stability margin.

Torque distribution optimization of redundantly actuated planar parallel mechanisms based on a null-space solution

SUMMARYRedundant actuation for the parallel kinematic machine (PKM) is a well-known technique for overcoming general drawbacks of the PKM by helping it to avoid singularity and enhance stiffness characteristics, among others. Torque distribution plays a critical role in redundant actuation because this actuation causes the PKM to consume too much energy or put a substantial amount of stress on joints and links. This paper proposes a new torque distribution method for reducing the maximum torque of the actuator of a planar PKM. Here the main idea behind the proposed method is the use of superposition of a particular solution for a non-redundant case and an optimized null-space solution for a redundant case with a constant coefficient. The optimal value of a null-space solution can be easily determined by checking only the intersection points of the profile of the actuator's torque as the coefficient varies. We consider three cases of planar PKMs—2-, 3-, and 4-RRR PKMs—and present a detailed procedure for deriving a kinematic solution for the 2-RRR PKM based on Screw theory. We compare the proposed method with the minimum-norm pseudo-inverse method and assess a limitation of the proposed method. The torque distribution algorithm can be used to determine the number of actuators in an efficient manner and to reduce energy consumption.

Adaptive Matrix/Vector Gradient Algorithm for Design of IIR Filters and ARMA Models

This work describes a novel adaptive matrix/vector gradient (AMVG) algorithm for design of IIR filters and ARMA signal models. The AMVG algorithm can track to IIR filters and ARMA systems having poles also outside the unit circle. The time reversed filtering procedure was used to treat the unstable conditions. The SVD-based null space solution was used for the initialization of the AMVG algorithm. We demonstrate the feasibility of the method by designing a digital phase shifter, which adapts to complex frequency carriers in the presence of noise. We implement the half-sample delay filter and describe the envelope detector based on the Hilbert transform filter.

Constructive solid analysis: a hierarchical, geometry-based meshless analysis procedure for integrated design and analysis

In this paper, we propose an analysis methodology that is procedurally analogous to Constructive Solid Geometry (CSG) integrating design and analysis, and thereby enabling efficient optimal design. The procedure, due to its analogous nature to CSG, is termed Constructive Solid Analysis (CSA). The analysis methodology is partitioned, hierarchical and is based on constructing the boundary value problem for a compound geometry through operations on the field quantities defined on the primitives. Although the CSA procedure will allow any basis for approximating the fields, Non-Uniform Rational B-Splines (NURBS), currently popular in the geometric modeling literature, are used to represent the geometry of the primitives as well as the analysis fields. The use of the same basis to represent geometry and analysis fields enables ‘representational’ integration, and further, the developed methodology may be classified as a partition of unity meshless analysis scheme. A more general null-space solution scheme and a somewhat restrictive range-space solution scheme are outlined to solve the discretized equations resulting from the use of NURBS. Several representative problems from the field of linear elasticity are solved to demonstrate the validity of the procedure and to evaluate its computational cost relative to the finite element method. The optimal orientation of an elliptical hole to applied tractions is determined to demonstrate the power of the proposed methodology for shape optimal design.

A Null-Space Solution of the Inverse Kinematics of Redundant Manipulators Based on a Decomposition of Screws

In this paper, an approach for solving the inverse kinematics (velocity) problem for redundant open-chain manipulators is addressed. The solution is based upon finding a particular joint rate solution and identifying a joint velocity null-space basis through a decomposition of screw coordinates representing the joint axes. Optimal joint rate solutions are formulated in terms of the joint null-space basis. A specification of computational costs for numerical application is presented. The method is shown to be useful for analytical application through example derivations for two 7 revolute jointed manipulators and a platform mounted nonredundant manipulator.