A quotient framework for single-input pole placement and associated algorithms

Single-input eigenvalue assignment problem (SEVAS) for dense (non-sparse) weakly controllable pairs A,B using Ackermann's formula is revisited. Factorizations are presented that are interpreted using quotient (factor) vector spaces. Depending on the representations chosen for the equivalence class (given as specific projections by real orthogonal matrices), the numerical behavior of pole placement scheme can be enhanced. One version operates by placing one pole at a time (with complex conjugate poles grouped together to avoid complex arithmetic). Another version operates with the coefficients of the characteristic polynomial directly. The latter version uses orthogoanl real matrices. In both cases, there are no constraint on the number of identical poles. A version of the algorithm uses only ring arithmetic. The algorithms are compared with numerically stable algorithms that appeared in the literature, such as the Miminis-Paige algorithm or the Varga pole shifting method. In case of real distinct eigenvalues to be placed, a geometrical interpretation as the interesection of affine hyperplanes provides the value of the gain vector, which seems a novel interpretation.Comment: 93 pages, 9 figure

On two isomorphic Lie algebroids for Feedback Linearization

Two Lie algebroids are presented that are linked to the construction of the linearizing output of an affine in the input nonlinear system.\ The algorithmic construction of the linearizing output proceeds inductively, and each stage has two structures, namely a codimension one foliation defined through an integrable 1-form $\omega$ , and a transversal vectorfield $g$\ to the foliation. Each integral manifold of the vectorfield $g$ defines an equivalence class of points. Due to transversality, a leaf of the foliation is chosen to represent these equivalence classes. A Lie groupoid is defined with its base given as the particular chosen leaf and with the product induced by the pseudogroup of diffeomorphisms that preserve equivalence classes generated by the integral manifolds of g. Two Lie algebroids associated with this groupoid are then defined. The theory is illustrated with an example using polynomial automorphisms as particular cases of diffeomorphisms and shows the relation with the Jacobian conjecture

Application of Artificial Neural Networks in Assessing the Equilibrium Depth of Local Scour Around Bridge Piers

Scour can have the effect of subsidence of the piers in bridges, which can ultimately lead to the total collapse of these systems. Effective bridge design needs appropriate information on the equilibrium depth of local scour. The flow field around bridge piers is complex so that deriving a theoretical model for predicting the exact equilibrium depth of local scour seems to be near impossible. On the other hand, the assessment of empirical models highly depends on local conditions, which is usually too conservative. In the present study, artificial neural networks are used to estimate the equilibrium depth of the local scour around bridge piers. Assuming such equilibrium depth is a function of five vari- ables, and using experimental data, a neural network model is trained to predict this equilibrium depth. Multilayer neural net- works with backpropagation algorithm with different learning rules are investigated and implemented. Different methods of data normalization besides the effect of initial weightings and overtraining phenomenon are addressed. The results show well adoption of the neural network predictions against experimental data in comparison with the estimation of empirical models

Time-projection control to recover inter-sample disturbances, application to bipedal walking control

We present a new walking controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of 3LP, the proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. The projection controller reacts to disturbances immediately and compared to the discrete LQR controller, it provides superior performance in recovering intermittent external pushes. Further analysis of closed-loop eigenvalues and disturbance rejection strength show strong stabilization properties for this architecture. An analysis of viable regions also show that the proposed controller covers most of the maximal viable set of states. It is computationally much faster than Model Predictive Controllers (MPC) and yet optimal over an infinite horizon

Imprecise dynamic walking with time-projection control

We present a new walking foot-placement controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of the 3LP model, our proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. This control paradigm reacts to disturbances immediately and includes rules to account for swing dynamics and leg-retraction. We apply it to a simulated Atlas robot in position-control, always commanded to perform in-place walking. The stance hip joint in our robot keeps the torso upright to let the robot naturally fall, and the swing hip joint tracks the desired footstep location. Combined with simple Center of Pressure (CoP) damping rules in the low-level controller, our foot-placement enables the robot to recover from strong pushes and produce periodic walking gaits when subject to persistent sources of disturbance, externally or internally. These gaits are imprecise, i.e., emergent from asymmetry sources rather than precisely imposing a desired velocity to the robot. Also in extreme conditions, restricting linearity assumptions of the 3LP model are often violated, but the system remains robust in our simulations. An extensive analysis of closed-loop eigenvalues, viable regions and sensitivity to push timings further demonstrate the strengths of our simple controller

Avoiding Feedback-Linearization Singularity Using a Quotient Method -- The Field-Controlled DC Motor Case

Feedback linearization requires a unique feedback law and a unique diffeomorphism to bring a system to Brunovsk´y normal form. Unfortunately, singularities might arise both in the feedback law and in the diffeomorphism. This paper demonstrates the ability of a quotient method to avoid or mitigate the singularities that typically arise with feedback linearization. The quotient method does it by relaxing the conditions on diffeomorphism, which can be achieved since there is an additional degree of freedom at each step of the iterative procedure. This freedom in choosing quotients and the resulting advantage are demonstrated for a field-controlled DC motor. Using a Lyapunov function, the domain of attraction of the control law obtained with the quotient method is proved to be larger than the domain of attraction of a control law obtained using feedback linearization

Numerical algorithm for feedback linearizable systems

A numerical algorithm that achieves asymptotic stability for feedback linearizable systems is presented. The nonlinear systems can be represented in various forms that include differential equations, simulated physical models or lookup tables. The proposed algorithm is based on a quotient method and proceeds iteratively. At each step, the dynamic system is desensitized with respect to the current input vector field. Control is obtained by tracking a desired value along the input vector field at each step. The numerical algorithm uses the direction on the tangent manifold at a given point and its variation around that point. This enables the algorithm to produce control values simply using a simulator of the nonlinear system

A quotient method for designing nonlinear controllers

An algorithmic method is proposed to design stabilizing control laws for a class of nonlinear systems that comprises single-input feedback-linearizable systems and a particular set of single-input non feedback-linearizable systems. The method proceeds iteratively and consists of two stages; it converts the system into cascade form and reduces the dimension at every step by creating quotient manifold in the forward stage, while it constructs the feedback law iteratively in the backward stage. The paper shows that the construction of these quotient manifolds is well defined for feedback-linearizable system and, furthermore, it can also be applied to a class of non feedback-linearizable systems

A framework for forward-dynamics simulation of the human shoulder

A vast majority of the available biomechanical models of the human shoulder has been developed based on inverse dynamics, e.g. [1,2]. This imposes a number of limitations on their application. For instance, the glenohumeral joint is approximated as an ideal joint in an inverse-dynamics simulation. Therefore, the models fall short to predict the joint translations [3]. The different approaches developed to overcome the recurrent limitations of the models can be broadly divided in two categories. The first category tries to tailor an available inverse-dynamics model to a specific application, e.g. [3,4]. The second category aims to develop a framework allowing forward-dynamics simulation, e.g. [5,6]. Indeed, few studies have developed forward-dynamics simulations of the human body. In [5], dynamic optimization was used to develop a forward-dynamics model of the lower extremity. Dynamic optimization typically demands many times integration of the equations of motion. Given the computational expense incurred by the integrations, the method is impractical for common applications. In this study, a framework for forward-dynamics simulation of the human shoulder is developed. In contrast with the dynamic optimization, the developed framework requires a single integration of the system equations. It is based on a joint application of a biomechanical model of the shoulder and a controller. The controller defines the muscle forces allowing the model to be simulated in forward dynamics. Different control scenarios are considered to investigate the model convergence in terms of accuracy and computational effort