Time-optimal Solution Research Articles

Robust Time-OptimaI Control for Flexible StructureVibration Control by Augmented Dynamics

Robust optimal control problem for flexible spacecraft/structures using on-off type discrete actuators has been a subject of intensive research. Optimization by switching time parameterization can be used to minimize the maneuver time subject to equality boundary constraints. Sensitivity of the switching times with respect to modal parameters could be a critical factor degrading the performance of the time optimal solutions. A new approach for the robustness enhancement of switching times with respect to modal parameter uncertainty is introduced. A new concept so-called augmented dynamic model is added to account for the modal uncertainty. The proposed approach turns out to be similar to some previous methods, but it essentially provides some new results.

Time-optimal multidimensional gradient waveform design for rapid imaging.

Magnetic resonance imaging (MRI) is limited in many cases by long scan times and low spatial resolution. Recent advances in gradient systems hardware allow very rapid imaging sequences, such as steady-state free precession (SSFP), which has repetition times (TRs) of 2-5 ms. The design of these rapid sequences demands time-optimal preparatory gradient waveforms to provide maximum readout duty-cycle, and preserve spatial resolution and SNR while keeping TRs low. Time-optimal gradient waveforms can be synthesized analytically for certain simple cases. However, certain problems, such as time-optimal 2D and 3D gradient design with moment constraints, either may not have a solution or must be solved numerically. We show that time-optimal gradient design is a convex-optimization problem, for which very efficient solution methods exist. These methods can be applied to solve gradient design problems for oblique gradient design, spiral imaging, and flow-encoding using either a constant slew rate or the more exact voltage-limited gradient models. Ultimately, these methods provide a time-optimal solution to many 2D and 3D gradient design problems in a sufficiently short time for interactive imaging.

Time-optimal control of a single-DOF mechanical system considering actuator dynamics

In this paper, we consider time-optimal control of a single degree-of-freedom (DOF) mechanical system in which a linear brushless dc motor (BLDCM) is used as the actuator. The time-optimal solution is obtained in feedback form via three-dimensional (3-D) state-space analysis. In its derivation, we take into full account the actuator dynamics as well as the physical limitations on current, voltage, and jerk. We also improve the well-known s-curve motion profiling so that both the actuator dynamics and the physical limitations are taken into full account. Through some experimental results, we demonstrate that the time-optimal motion profiling still can provide faster trajectories for motion planning than the improved s-curve motion profiling.

Nearly time-optimal paths for a ground vehicle

It is well known that the sufficient family of time-optimal paths for both Dubins' as well as Reeds-Shepp' s car models consist of the concatenation of circular arcs with maximum curvature and straight line segments, all tangentially connected. These time-optimal solutions suffer from some drawbacks. Their discontinuous curvature profile, together with the wear and impairment on the control equipment that the bang-bang solutions induce, calls for ' smoother and more supple reference paths to follow. Avoiding the bang-bang solutions also raises the robustness with respect to any possible uncertainties. In this paper, our main tool for generating these “nearly time-optimal” , but nevertheless continuous-curvature paths, is to use the Pontryagin Maximum Principle (PMP) and make an appropriate and cunning choice of the Lagrangian function. Despite some rewarding simulation results, this concept turns out to be numerically divergent at some instances. Upon a more careful investigation, it can be concluded that the problem at hand is nearly singular. This is seen by applying the PMP to Dubins car and studying the corresponding two point boundary value problem, which turn out to be singular. Realizing this, one is able to contradict the widespread belief that all the information about the motion of a mobile platform lies in the initial values of the auxiliary variables associated with the PMP.

A phase-plane approach to time-optimal control of single-DOF mechanical systems with friction

In this paper, we attempt to solve the time-optimal control problem for single-degree-of-freedom (DOF) mechanical systems with friction, while taking into account not only the velocity-dependent control input constraint but also the state constraint. Direct application of the Pontryagin's maximum principle (PMP) leads to a sixth-order nonlinear two-point boundary-value problem (TPBVP) which is very difficult to solve numerically. In this context, we take a phase-plane analysis without resorting to the PMP. Thereby, the exact time-optimal solution can be obtained simply by solving a set of first-order differential equations with continuous right-hand sides. We also present some simulation results to demonstrate the practical use of the time-optimal solution.

The undershoot bias: learning to act optimally under uncertainty.

Learning in stochastic environments is increasingly viewed as an important psychological ability. To extend these results from a perceptual to a motor domain, we tested whether participants could learn to solve a stochastic minimal-time task using exploratory learning. The task involved moving a cursor on a computer screen to a target. We systematically varied the degree of random error in movement in three different conditions; each condition had a distinct time-optimal solution. We found that participants approximated the optimal solutions with practice. The results show that adults are sensitive to the stochastic structure of a task and naturally adjust the magnitude of an undershoot bias to the particular movement error of a task.

A time-optimal solution for the path cover problem on cographs

We show that the notoriously difficult problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved time- and work-optimally for cographs. Our result implies that for this class of graphs the task of finding a Hamiltonian path can be solved time- and work-optimally in parallel.It was open for more than 10 years to find a time- and work-optimal parallel solution for this important problem. Our contribution is to offer an optimal solution to this important problem. We begin by showing that any algorithm that solves an instance of size n of the problem must take Ω(logn) time on the CREW, even if an infinite number of processors are available. We then go on to show that this time lower bound is tight by devising an EREW algorithm that, given an n-vertex cograph G represented by its cotree, finds and reports all the paths in a minimum path cover in O(logn) time using n/logn processors.

Analytical time optimal control solution for a two-link planar aerobot with initial angular momentum

In order to control gymnastic and jumping robots, we derive the complete analytical solution to the posture control problem of a two-link free flying object with initial angular momentum. We show that the solution involves singular control and derive formulas to calculate the optimal switching condition, optimal terminal time and optimal trajectories. As an application, a high diving motion is simulated.

Leaf sequencing with secondary beam blocking under leaf positioning constraints for continuously modulated radiotherapy beams

The creation of arbitrary photon fluence patterns for intensity modulated radiotherapy is addressed. The proposed method is intended for a class of multileaf collimators with a requirement for minimum leaf separation. Unlike the solution of Convery and Webb in which discrete beam intensity modulation was assumed, the present method deals with continuous modulation or that consisting of infinitely small bixels. The method begins with the time-optimal solution of Spirou-Stein-Svensson disregarding the minimum gap requirement. Subsequently, the gaps are restored by mobilizing the secondary beam blocking devices to prevent overexposure resulting from the leaf separation process. The secondary beam blocking is provided by means of two orthogonal backup diaphragms that are computer controlled. The results indicate that the method can be used to accurately deliver the desired modulation while satisfying the leaf positioning constraints. Furthermore, an example is presented which illustrates the efficacy of using the horizontal backup diaphragms (moving in perpendicular direction of the leaves) in addition to the vertical backup diaphragms (moving in the parallel direction of the leaves) to generate zero fluence regions.

On time optimal solutions of the firing squad synchronization problem for two-dimensional paths

The firing squad synchronization problem for two-dimensional paths (2-Path FSSP, for short) is a variation of the firing squad synchronization problem where finite automata are placed along a path in the two-dimensional array space. Whether 2-Path FSSP has a time optimal solution or not is an open problem. We introduce a combinatorial problem which we call the two-dimensional path extension problem (2-PEP, for short), and show that if 2-Path FSSP has a time optimal solution then 2-PEP has a polynomial time algorithm. The computational complexity of 2-PEP is not well understood and the exhaustive search requiring exponential time is the only algorithm we know for it at present.

Optimal total exchange in Cayley graphs

Consider an interconnection network and the following situation: Every node needs to send a different message to every other node. This is the total exchange or all-to-all personalized communication problem, one of a number of information dissemination problems known as collective communications. Under the assumption that a node can send and receive only one message at each step (single-port model), it is seen that the minimum time required to solve the problem is governed by the status (or total distance) of the nodes in the network. We present a time-optimal solution for any Cayley network. Rings, hypercubes, cube-connected cycles, and butterflies are some well-known Cayley networks which can take advantage of our method. The solution is based on a class of algorithms which we call node-invariant algorithms and which behave uniformly across the network.

Minimum Time Trajectory Optimization and Learning

This paper presents a numerical algorithm for finding the bang-bang control input associated with the time optimal solution of a class of nonlinear dynamic systems. The proposed algorithm directly searches for the optimal switching instants based on a projected gradient optimization method. It is shown that this algorithm can be made into a learning algorithm by using on-line measurements of the state trajectory. The learning is shown to have the potential for significant robustness to mismatch between the model and the system. It learns a nearly optimal input through repeated trials in which it utilizes the measured terminal state error of the actual system and gradients based on the theoretical state equation of the system but evaluated along the actual state trajectory. The success of the method is demonstrated on an underactuated double pendulum system called the acrobot.

Creating Time-Optimal Commands with Practical Constraints

In applications that demand rapid response, time-optimal control techniques are often enlisted. Recently, a new technique has been presented for deriving time-optimal command proe les by solving a set of algebraic constraint equations. This time-optimal solution framework is shown to be easily extendable to derive commands satisfying a variety of practical constraints, particularly constraints on command length and sensitivity to modeling errors.

Cheap Newton steps for optimal control problems: automatic differentiation and Pantoja's algorithm

In this paper we discuss Pantoja's construction of the Newton direction for discrete time optimal control problems. We show that automatic differentiation (AD) techniques can be used to calculate the Newton direction accurately, without requiring extensive re-writing of user code, and at a surprisingly low computational cost: for an N-step problem with p control variables and q state variables at each step, the worst case cost is 6(p + q + 1) times the computational cost of a single target function evaluation, independent of N, together with at most p 3/3 + p 2(q + 1) + 2p(q + 1)2 + (q + l)3, i.e. less than (p + q + l)3, floating point multiply-and-add operations per time step. These costs may be considerably reduced if there is significant structural sparsity in the problem dynamics. The systematic use of checkpointing roughly doubles the operation counts, but reduces the total space cost to the order of 4pN floating point stores. A naive approach to finding the Newton step would require the solution of an Np × Np system of equations together with a number of function evaluations proportional to Np, so this approach to Pantoja's construction is extremely attractive, especially if q is very small relative to N. Straightforward modifications of the AD algorithms proposed here can be used to implement other discrete time optimal control solution techniques, such as differential dynamic programming (DDP), which use state-control feedback. The same techniques also can be used to determine with certainty, at the cost of a single Newton direction calculation, whether or not the Hessian of the target function is sufficiently positive definite at a point of interest. This allows computationally cheap post-hoc verification that a second-order minimum has been reached to a given accuracy, regardless of what method has been used to obtain it.

Determination of Minimum-Time Maneuvers for a Robotic Manipulator Using Numerical Optimization Methods*

The productivity of automated production lines depends on the velocities of operating robot manipulators. Hence, time-optimal control for working cycles of robot manipulators are of decisive importance. In this paper, the minimum-time retraction of a robot arm, subject to gravity and suspended on a prismatic-rotational joint, is investigated. Iterative integration methods using a B-spline representation for the actuator force and torque, a Runge-Kutta integration method, and a sequential quadratic programming optimization algorithm are used to calculate the time-optimal control and trajectories. Results show that nonlinear gravity and centrifugal effects are exploited very effectively, to obtain minimum-time maneuvers from an initial to a final state. These states also determine the switching structures of the control. It is demonstrated that even the simple retraction of a robot arm produces unexpectedly complex time-optimal solutions.

An Algorithm for Time-Optimal Control Problems

A numerical procedure to compute nonsingular, time-optimal solutions for nonlinear systems, which have fixed initial and final states, and are linear and bounded in control, is presented. Using the Pontryagin’s Minimum Principle, the corresponding nonlinear two-point boundary-value problem is formulated and solved by combination of the forward-backward and the shooting methods. The forward-backward procedure generates a good guess of the initial costates, which is crucial for the convergence of the shooting method. Numerical example of a two-link manipulator illustrates the proposed approach and the convergence of the procedure.

Robust minimum time control of flexible structures

We investigate the existence and uniqueness of robust time-optimal control solutions of flexible structures for rest-to-rest motion. We show that several robust time-optimal control designs for flexible systems are equivalent to traditional nonrobust time-optimal controls for different systems. This equivalence implies that the robust time-optimal controllers must then have all the properties that are known about traditional timeoptimal controllers. Further, many of the numerical methods developed for generating time-optimal commands are directly applicable to designing robust time-optimal commands.

A time-optimal solution to a classification problem in ordered functional domains, with applications

Consider a family C of classifiers represented by continuous functions stored, in discretized form, one function per processor in one column of a mesh with multiple broadcasting of size √ n × √ n In a number of contexts in pattern recognition, morphology, and knowledge engineering applications, it is necessary to answer point location queries with respect to the given classifiers. The main contribution of this work is to show that in such a domain a collection of m (1 ⪕ m ⪕ n), point location (i.e. classification) queries can be answered in Θ( m/ √ m) time. As it turns out, this is best possible on this platform.

Assembly scheduling for an integrated two-robot workcell

An assembly sequence planning scheme, giving the time optimal solution for a two-robot cell, is presented. The scheme is based on dynamic programming, and the time optimum assembly sequence is generated in two stages. In stage one, an initial candidate sequence is derived. The stage-one algorithm is computationally efficient, the complexity being O( log n k ) , where n is the number of elements in k groups. It also gives near-optimal results. Expression for the lower bounds for the total assembly time are derived in order to assess the departure from optimality of the stage-one sequence. In stage two, the initial candidate sequence is optimized using an iterative technique to yield the time optimal sequence. The scheme is extended to account for precedence constraints. It can also be extended for assembly cells with more than two robots. The scheme is tested using computer simulations, and some test results are presented. For the problems solved (typically involving 20 elements), the computational times were less than 5 sec in an Apollo 300 workstation.

On Singular Phenomena in Certain Time-Optimal Problem

The purpose of this paper is to present the solution of time-optimal problem of the controlled object the dynamics of which is given by: \dot{x}=y,\dot{y}=f(x)+u , where |u|\leq1 and motion resistance function f(x)=0 ifx\leq 0 , f(x)=-A if x>0 where 0\leq A A_b we will examine two following singular phenomena. (a) If the target state \vec{z}_1=(0, 0) then there exists the switching curve, dividing the state plane into two sets, however only one its branch is formed by the time-optimal solution reaching the target\vec{z}_1=(0, 0) and generated by the control u=-1. None of solution forms the second branch of switching curve. It is formed by a state-locus depending on the value of A only. In dependency of the starting state \vec{z}_0 the time-optimal control process is generated by bang-bang control with none, one or two switching operations. This is the first singular phenomenon, because any small decrease of the value A over A_b requires to change the structure which would be able to generate the time-optimal process. (b) The paper shows, that if the target state \vec{z}_1(x_1, 0), x_1>0 then there exists a set of the starting states from which there start two trajectories reaching the target in the same minimum time. This is the second phenomenon. Finally, some suggestions as to practical applications have been given too.