Smooth Approximation Research Articles

Hybrid UWB-Inertial TDoA-Based Target Tracking With Concentrated Anchors

In this paper, hybrid radio/inertial mobile target tracking for accurate and smooth path estimation is considered. The proposed tracking approach builds upon an Ultra WideBand (UWB)-based positioning algorithm, based on the Linear Hyperbolic Positioning System (LinHPS), with Time Difference of Arrival (TDoA) processing and anchors concentrated on a single hotspot at the center of the environment where the target moves. First, we design an Adaptive Radio-based Extended Kalman Filter (AREKF), which does not require a priori statistical knowledge of the noise in the target movement model and estimates the measurement noise covariance, at each sampling time, according to a proper LookUp Table (LUT). In order to improve the performance of AREKF, we incorporate inertial data collected from the target and propose three “hybrid” radio/inertial algorithms, denoted as Hybrid Inertial Measurement Unit (IMU)-aided Radio-based EKF (HIREKF), Hybrid Noisy Control EKF (HNCEKF), and Hybrid Control EKF (HCEKF). Our results on experimentally acquired paths show that the proposed algorithms achieve an average instantaneous position estimation error on the order of a few centimeters. Moreover, the minimum target path length estimation error, obtained with HCEKF, is on the order of 6% and 1% for two paths with lengths equal to approximately 17 m and 46 m, respectively.

Smooth Uniform Approximation of Multivariate Spline Function

Multivariable spline approximation is of great significance both in mathematics and in practice. This involves the approximation of non-smooth functions. In this paper, the approximation of the non-smooth function , which is the most representative problem in robust statistics, is discussed. We propose a compact smooth uniform approximation algorithm using spline curves.

APPROXIMATIONS OF THE THEORIES OF STRUCTURES WITH ONE EQUIVALENCE RELATION

Recently, various methods similar to the “transfer principle” have been rapidly developing, where one property of a structure or pieces of this structure is satisfied in all infinite structures or in another algebraic structure. Such methods include smoothly approximable structures, holographic structures, almost sure theories, and pseudofinite structures approximable by finite structures. Pseudofinite structures are mathematical structures that resemble finite structures but are not actually finite. They are important in various areas of mathematics, including model theory and algebraic geometry. Pseudofinite structures are a fascinating area of mathematical logic that bridge the gap between finite and infinite structures. They allow studying infinite structures in ways that resemble finite structures, and they provide a connection to various other concepts in model theory. Further studying pseudofinite structures will continue to reveal new insights and applications in mathematics and beyond. Pseudofinite theory is a branch of mathematical logic that studies structures that are similar in some ways to finite structures, but can be infinitely large in other ways. It is an area of research that lies at the intersection of model theory and number theory and deals with infinite structures that share some properties with finite structures, such as having only finitely many elements up to isomorphism. A. Lachlan introduced the concept of smoothly approximable structures in order to change the direction of analysis from finite to infinite, that is, to classify large finite structures that seem to be smooth approximations to an infinite limit. The theory of pseudofinite structures is particularly relevant for studying equivalence relations. In this paper, we study the model-theoretic property of the theory of equivalence relations, in particular, the property of smooth approximability. Let L = {E}, where Е is an equivalence relation. We prove that an any ω-categorical L-structure M is smoothly approximable. We also prove that any infinite L-structure M is pseudofinite.

Learning Observation Model for Factor Graph Based-State Estimation Using Intrinsic Sensors

The navigation system of autonomous mobile robots has appeared challenging when using exteroceptive sensors such as cameras, LiDARs, and radars in textureless and structureless environments. This paper presents a robust state estimation system for holonomic mobile robots using intrinsic sensors based on adaptive factor graph optimization in the degradation scenarios. In particular, the neural networks are employed to learn the observation and noise model using only IMU sensor and wheel encoder data. Investigating the learning model for the holonomic mobile robot is discussed with various neural network architectures. We also explore the neural networks that are far more powerful and have cheaper computing power when using the inertial-wheel encoder sensors. Furthermore, we employ an industrial holonomic robot platform equipped with multiple LiDARs, cameras, IMU, and wheel encoders to conduct the experiments and create the ground truth without a bulky motion capture system. The collected datasets are then implemented to train the neural networks. Finally, the experimental evaluation presents that our solution provides better accuracy and real-time performance than other solutions. <i>Note to Practitioners</i>&#x2014;Autonomous mobile robots need to serve in challenging environments robustly that deny extrinsic sensors such as cameras, LiDARs, and radars. In order to operate in this situation, the navigation system shall rely on the intrinsic sensor as inertial sensor and wheel encoders. Existing conventional methods have combined the intrinsic sensors in the form of the recursive Bayesian filtering technique without adapting these models. Besides, deep learning-based solutions have adopted extensive networks like LSTM or CNN to handle the estimation problem. This work aims to develop a state estimation subsystem of the navigation system for the holonomic mobile robots that utilize the intrinsic sensors in adaptive factor graph optimization. In particular, we present how to join the factor graph efficiently with the learning observation model of IMU and wheel encoder factor. Moreover, the neural networks are introduced to learn the observation model with an IMU and wheel encoder data inputs. We recognize that lightweight neural networks can achieve more power than deep learning techniques using the IMU sensor and wheel encoders. Finally, the neural networks are embedded in a factor graph to handle the smoothing state estimation. The proposed system could operate with high accuracy in real time.

Dynamical BLUP modeling of reaction norm evolution, accommodating changing environments, overlapping generations, and multivariate data

AbstractFor theoretical studies, reaction norm evolution in a changing environment can be modeled by means of the multivariate breeder's equation, with the reaction norm parameters treated as traits in their own right. This is, however, not a feasible approach for the use of field data, where the intercept and slope values are not available. An alternative approach is to use infinite‐dimensional characters and smooth covariance function estimates found by, e.g., random regression. This is difficult because of the need to find, for example, polynomial basis functions that fit the data reasonably well over time, and because reaction norms in multivariate cases are correlated, such that they cannot be modeled independently. Here, I present an alternative approach based on a multivariate linear mixed model of any order, with dynamical incidence and residual covariance matrices that reflect the changing environment. From such a mixed model follows a dynamical BLUP model for the estimation of the individual reaction norm parameter values at any given parent generation, and for updating of the mean reaction norm parameter values from generation to generation by means of Robertson's secondary theorem of natural selection. This will, for example, make it possible to disentangle the microevolutionary and plasticity components in climate change responses. The BLUP model incorporates the additive genetic relationship matrix in the usual way, and overlapping generations can easily be accommodated. Additive genetic and environmental model parameters are assumed to be known and constant, but it is discussed how they can be estimated by means of a prediction error method. The identifiability by the use of field or laboratory data containing environmental, phenotypic, fitness, and additive genetic relationship data is an important feature of the proposed model.

Secrecy Rate Maximization of RIS-Assisted SWIPT Systems: A Two-Timescale Beamforming Design Approach

Reconfigurable intelligent surfaces (RISs) achieve high passive beamforming gains for signal enhancement or interference nulling by dynamically adjusting their reflection coefficients. Their employment is particularly appealing for improving both the wireless security and the efficiency of radio frequency (RF)-based wireless power transfer. Motivated by this, we conceive and investigate a RIS-assisted secure simultaneous wireless information and power transfer (SWIPT) system designed for information and power transfer from a base station (BS) to an information user (IU) and to multiple energy users (EUs), respectively. Moreover, the EUs are also potential eavesdroppers that may overhear the communication between the BS and IU. We adopt <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">two-timescale</i> transmission for reducing the signal processing complexity as well as channel training overhead, and aim for maximizing the average worst-case secrecy rate achieved by the IU. This is achieved by jointly optimizing the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">short-term</i> transmit beamforming vectors at the BS (including information and energy beams) as well as the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">long-term</i> phase shifts at the RIS, under the energy harvesting constraints considered at the EUs and the power constraint at the BS. The stochastic optimization problem formulated is non-convex with intricately coupled variables, and is non-smooth due to the existence of multiple EUs/eavesdroppers. No standard optimization approach is available for this challenging scenario. To tackle this challenge, we propose a smooth approximation aided stochastic successive convex approximation (SA-SSCA) algorithm. Furthermore, a low-complexity heuristic algorithm is proposed for reducing the computational complexity without unduly eroding the performance. Simulation results show the efficiency of the RIS in securing SWIPT systems. The significant performance gains achieved by our proposed algorithms over the relevant benchmark schemes are also demonstrated.

The impossible challenge of estimating non-existent moments of the Chemical Master Equation.

The Chemical Master Equation (CME) is a set of linear differential equations that describes the evolution of the probability distribution on all possible configurations of a (bio-)chemical reaction system. Since the number of configurations and therefore the dimension of the CME rapidly increases with the number of molecules, its applicability is restricted to small systems. A widely applied remedy for this challenge is moment-based approaches which consider the evolution of the first few moments of the distribution as summary statistics for the complete distribution. Here, we investigate the performance of two moment-estimation methods for reaction systems whose equilibrium distributions encounter fat-tailedness and do not possess statistical moments. We show that estimation via stochastic simulation algorithm (SSA) trajectories lose consistency over time and estimated moment values span a wide range of values even for large sample sizes. In comparison, the method of moments returns smooth moment estimates but is not able to indicate the non-existence of the allegedly predicted moments. We furthermore analyze the negative effect of a CME solution's fat-tailedness on SSA run times and explain inherent difficulties. While moment-estimation techniques are a commonly applied tool in the simulation of (bio-)chemical reaction networks, we conclude that they should be used with care, as neither the system definition nor the moment-estimation techniques themselves reliably indicate the potential fat-tailedness of the CME's solution.

Random data final-state problem of fourth-order inhomogeneous NLS

We consider the fourth-order inhomogeneous nonlinear Schrödinger equation with mass-subcritical nonlinearity iut+Δ2u+λ|x|−α|u|βu=0. For u+∈L2(Rd), we perform a physical-space randomization and get a random final state u+ω∈L2(Rd). By establishing the smoothing estimate, time-space-weighted inhomogeneous Strichartz estimate and random Strichartz estimate, we prove that for almost every ω, there exists a unique, global solution to the fourth-order inhomogeneous nonlinear Schrödinger equation that scatters to u+ω∈L2.

State of charge estimation for lithium-ion batteries based on cross-domain transfer learning with feedback mechanism

When the deep learning model is applied to estimate battery state of charge (SOC), the information inside the training set cannot be leveraged thoroughly, which would cause poor SOC estimation accuracy and robustness on the testing set. To solve the problem, this paper proposes an adaptive convolutional neural network-gated recurrent unit with Kalman filter and feedback mechanism (Fb-Ada-CNN-GRU-KF) for SOC estimation considering distribution difference of data segments inside the training set through transfer learning and extracting the spatial information through convolutional layer. Furthermore, the feedback mechanism provides the model more information to learn to correct the systematic error, and the KF in the proposed model works as a post data processor to obtain a steady and smooth SOC estimation results. Experimental and comparison results show that the proposed model for SOC estimation outperforms the existing deep learning methods in terms of the accuracy, generalization and stability.

Battery SOH estimation based on decision tree and improved support vector machine regression algorithm

Battery state of health (SOH) estimation is crucial for the estimation of the remaining driving range of electric vehicles and is one of the core functions of the battery management system (BMS). The lithium battery feature sample data used in this paper is extracted from the National Aeronautics and Space Administration (NASA) of the United States. Based on the obtained feature samples, a decision tree algorithm is used to analyze them and obtain the importance of each feature. Five groups of different feature inputs are constructed based on the cumulative feature importance, and the original support vector machine regression (SVR) algorithm is applied to perform SOH estimation simulation experiments on each group. The experimental results show that four battery features (voltage at SOC = 100%, voltage, discharge time, and SOC) can be used as input to achieve high estimation accuracy. To improve the training efficiency of the original SVR algorithm, an improved SVR algorithm is proposed, which optimizes the differentiability and solution method of the original SVR objective function. Since the loss function of the original SVR is non-differentiable, a smoothing function is introduced to approximate the loss function of the original SVR, and the original quadratic programming problem is transformed into a convex unconstrained minimization problem. The conjugate gradient algorithm is used to solve the smooth approximation objective function in a sequential minimal optimization manner. The improved SVR algorithm is applied to the simulation experiment with four battery feature inputs. The results show that the improved SVR algorithm significantly reduces the training time compared to the original SVR, with a slight trade-off in simulation accuracy.

Probability Density Estimation through Nonparametric Adaptive Partitioning and Stitching

We present a novel nonparametric adaptive partitioning and stitching (NAPS) algorithm to estimate a probability density function (PDF) of a single variable. Sampled data is partitioned into blocks using a branching tree algorithm that minimizes deviations from a uniform density within blocks of various sample sizes arranged in a staggered format. The block sizes are constructed to balance the load in parallel computing as the PDF for each block is independently estimated using the nonparametric maximum entropy method (NMEM) previously developed for automated high throughput analysis. Once all block PDFs are calculated, they are stitched together to provide a smooth estimate throughout the sample range. Each stitch is an averaging process over weight factors based on the estimated cumulative distribution function (CDF) and a complementary CDF that characterize how data from flanking blocks overlap. Benchmarks on synthetic data show that our PDF estimates are fast and accurate for sample sizes ranging from 29 to 227, across a diverse set of distributions that account for single and multi-modal distributions with heavy tails or singularities. We also generate estimates by replacing NMEM with kernel density estimation (KDE) within blocks. Our results indicate that NAPS(NMEM) is the best-performing method overall, while NAPS(KDE) improves estimates near boundaries compared to standard KDE.

On some non parametric estimators of the quantile density function for a stationary associated process

In this article, we consider smooth estimators for the quantile density function (qdf) for a sequence { X n , n ≥ 1 } of stationary non negative associated random variables with a common marginal distribution function. The qdf is given by q ( u ) = Q ′ ( u ) , u ∈ ( 0 , 1 ) , Q ( u ) representing the corresponding quantile function. The smooth estimators of q ( u ) considered here are adapted from those of Q ( u ) considered in Chaubey, Dewan, and Li (2021). A few asymptotic properties of these estimators are established parallel to those in the i.i.d. case. A numerical study comparing the mean squared errors of various estimators indicates the advantages and a few limitations of various estimators. The smoothing parameter is selected based on the BCV and RLCV (a variation of likelihood cross-validation) criteria. It is concluded, based on the numerical studies, that the RLCV criterion may produce over-smoothing, hence BCV criterion may be preferable. The numerical studies also suggest that, overall, the estimator proposed by Soni, Dewan, and Jain (2012) seems to have some advantage over the other estimators considered in this article.

A mark-specific quantile regression model.

Quantile regression has become a widely used tool for analysing competing risk data. However, quantile regression for competing risk data with a continuous mark is still scarce. The mark variable is an extension of cause of failure in a classical competing risk model where cause of failure is replaced by a continuous mark only observed at uncensored failure times. An example of the continuous mark variable is the genetic distance that measures dissimilarity between the infecting virus and the virus contained in the vaccine construct. In this article, we propose a novel mark-specific quantile regression model. The proposed estimation method borrows strength from data in a neighbourhood of a mark and is based on an induced smoothed estimation equation, which is very different from the existing methods for competing risk data with discrete causes. The asymptotic properties of the resulting estimators are established across mark and quantile continuums. In addition, a mark-specific quantile-type vaccine efficacy is proposed and its statistical inference procedures are developed. Simulation studies are conducted to evaluate the finite sample performances of the proposed estimation and hypothesis testing procedures. An application to the first HIV vaccine efficacy trial is provided.

Semiparametric estimation of latent variable asset pricing models

This paper studies semiparametric identification and estimation of consumption-based asset pricing models with latent state variables. First, we measure how consumption, dividends, and prices depend on Markovian state variables describing aggregate output growth. Subsequently, we identify state-dependent components in the stochastic discount factor using the Euler equation. We develop tractable algorithms for filtering, smoothing, and sieve maximum likelihood estimation, and establish its consistency. Empirically, we find sizable nonlinearities in the impact of expected growth and volatility on the price–dividend ratio and discount factor.

Framed Curves, Ribbons, and Parallel Transport on the Sphere

We consider curves γ:[0,1]→R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : [0, 1]\\rightarrow {\\mathbb {R}}^3$$\\end{document} endowed with an adapted orthonormal frame r:[0,1]→SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r: [0, 1]\\rightarrow SO(3)$$\\end{document}. We wish to deform such framed curves (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of r′rT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r'r^T$$\\end{document}), and a global constraint prescribing the initial and terminal values of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.

Underwater image restoration based on adaptive parameter optimization of the physical model.

Underwater images have the advantage of carrying high information density and are widely used for marine information acquisition. Due to the complex underwater environment, the captured images are often unsatisfactory and often suffer from color distortion, low contrast, and blurred details. Physical model-based methods are often used in relevant studies to obtain clear underwater images; however, water selectively absorbs light, making the use of a priori knowledge-based methods no longer applicable and thus rendering the restoration of underwater images ineffective. Therefore, this paper proposes an underwater image restoration method based on adaptive parameter optimization of the physical model. Firstly, an adaptive color constancy algorithm is designed to estimate the background light value of underwater image, which effectively guarantees the color and brightness of underwater image. Secondly, aiming at the problem of halo and edge blur in underwater images, a smoothness and uniformity transmittance estimation algorithm is proposed to make the estimated transmittance smooth and uniform, and eliminate the halo and blur of the image. Then, in order to further smooth the edge and texture details of the underwater image, a transmittance optimization algorithm for smoothing edge and texture details is proposed to make the obtained scene transmittance more natural. Finally, combined with the underwater image imaging model and histogram equalization algorithm, the image blurring is eliminated and more image details are retained. The qualitative and quantitative evaluation on the underwater image dataset (UIEBD) shows that the proposed method has obvious advantages in color restoration, contrast and comprehensive effect, and has achieved remarkable results in application testing. It shows that the proposed method can effectively restore underwater degraded images and provide a theoretical basis for the construction of underwater imaging models.

Three‐Dimensional Steady‐State Hydraulic Tomography Analysis With Integration of Cross‐Hole Flowmeter Data at a Highly Heterogeneous Site

AbstractHydraulic tomography (HT) has been shown to be a robust approach for the high‐resolution characterization of subsurface heterogeneity. However, HT can yield smooth estimates of hydraulic parameters when pumping tests and drawdown measurements are sparse, thus limiting the utility of characterization results in predicting groundwater flow and solute transport. To overcome this issue, this study integrates cross‐hole flowmeter measurements with HT analysis of steady‐state pumping/injection test data for the three‐dimensional (3‐D) characterization of hydraulic conductivity (K) at a highly heterogeneous glaciofluvial deposit site, which has not been previously attempted. Geostatistical inverse analyses of cross‐hole flowmeter data are conducted to yield preliminary estimates of K distribution, which are then utilized as initial K fields for steady‐state HT analysis of head data. Four cases combining three data types (geological information, cross‐hole flowmeter measurements, and steady‐state head data) for inverse modeling are performed. Model calibration and validation results from all cases are compared qualitatively and quantitatively to evaluate their performances. Results from this study show that (a) geostatistical inverse analysis of cross‐hole flowmeter data is capable in revealing vertical distributions of K at well locations and major high/low K zones between wells, (b) cross‐hole flowmeter data carry non‐redundant information of K heterogeneity compared to geological information and steady‐state head data, and (c) integration of flowmeter data improves characterization results in terms of revealing K heterogeneity details and predicting independent hydraulic test data. Therefore, this study demonstrates the usefulness of cross‐hole flowmeter data in augmenting HT surveys for improved K characterization in 3‐D.

A weighted independence test based on smooth estimation of Kendall distribution

In this study, we introduce a nonparametric test of independence which is based on a weighted Cramér-von Mises distance of the Bernstein estimate of the Kendall distribution function. We examine the power and the size of the new weighted test and, we compare it with the non-weighted nonparametric tests of independence through a Monte Carlo simulation study. The simulation results indicate that both of the choices of the weights and the polynomial degree has a significant effect on the test power. Finally, the procedure is applied to a dataset on chemical elements in water samples for the purpose of illustration.

Evolution of Esthetic Publications in Dentistry, Research Trends, and Global Productivity: A Bibliometric Analysis.

To perform a bibliometric study in the literature about the increasing number of global studies on dental esthetics in recent years. Articles on dental esthetics published between 1980 and 2021 were pulled from the Web of Science (WoS) database. In this comprehensive bibliometric study on dental esthetics, a summary of 2,858 articles published in the last 41 years was shared. Network visualization maps were used to identify citation analyses and trending topics. Spearman correlation coefficient was used for correlation studies. The exponential smoothing estimator was used to estimate the number of articles to be published in the coming years. Of the 3,666 publications, 2,858 (77.9%) were articles. The top three contributing countries to the literature were the USA (816, 28.5%), Brazil (286, 10%), and Italy (246, 8.6%). The most active institution was the University of Bern (n = 55). The journal with the highest number of articles was Journal of Prosthetic Dentistry (n = 178). According to the average number of citations per article, the most influential journal was Journal of Periodontology (citations: 55.6). Keywords of the trending topics related to dental esthetics are perception, bleaching, cleft lip, connective tissue graft, smile esthetics, adolescent, esthetic region, soft tissue augmentation, and clinical studies. In regard to the global output in dental esthetics, this article is a useful resource for clinicians, scientists, and dental students.

Fast Homotopy for Spacecraft Rendezvous Trajectory Optimization with Discrete Logic

This paper presents a computationally efficient optimization algorithm for solving nonconvex optimal control problems that involve discrete logic constraints. Traditional solution methods require binary variables and mixed-integer programming (MIP), which is prohibitively slow and computationally expensive. This paper proposes a faster and computationally cheaper algorithm that can produce locally optimal solutions in seconds. This is achieved by blending sequential convex programming and numerical continuation into a single iterative solution process. The algorithm approximates discrete logic constraints with smooth functions and uses a homotopy parameter to control the accuracy of this approximation. The homotopy parameter is updated such that, by the time the algorithm converges, the smooth approximations enforce the exact discrete logic. The effectiveness of this approach is numerically demonstrated for a realistic rendezvous scenario inspired by the Apollo Transposition and Docking maneuver. In less than 15 s of cumulative solver time, the algorithm finds a fuel-minimizing trajectory that obeys the following discrete logic constraints: thruster minimum impulse-bit, range-triggered approach cone, and range- triggered plume impingement. The optimized trajectory uses significantly less fuel than reported NASA design targets.