Piecewise Linear Trajectories Research Articles

Bayesian changepoint models for amyloid‐centric synchronisation of biomarker trajectories in AIBL

AbstractBackgroundThe modelling of biomarker dynamics in Alzheimer’s Disease from cohort studies faces challenges due to the lack of clear temporal points of references and the natural variability across individuals. Mixed‐effects models are often used to account for individual differences, but a disease timescale can enable better population‐level modelling than age or time since enrolment. Previous literature explored the temporal synchronisation of patients through observed time of conversion to MCI or AD, amyloid positivity, or aligning cognitive trajectories. We propose an amyloid‐centric synchronisation approach that estimates patient‐specific time‐shifts to best align them on a population‐level piecewise linear trajectory.MethodWe rely on 521 patients of the AIBL study with at least 3 measurements of PET amyloid, and model a 3‐segment, piecewise linear trajectory of centiloid over time. Our Bayesian model assumes that observed amyloid values are noisy realisations of the piecewise linear trajectory, and that the time of measurements is shifted by some individual‐specific lag. The first segment is constrained to have a zero slope, and individuals who were not observed to reach the first changepoint were excluded from subsequent analysis, as they may never develop Alzheimer’s Disease.ResultWe identified a changepoint in the amyloid trajectory at a median of (89% credible interval: [14.45, 23.64]) after initial accumulation in shifted time. Patients with measurements falling within this credible interval have a median of 18.9 centiloid (range of 2.0 to 30.1). On average, this changepoint describes an acceleration in amyloid accumulation, switching from a median 0.75 centiloid/year [0.65, 0.87] rate to 6.38 centiloid/year [6.20, 6.56] (Figure 1).ConclusionSynchronisation approaches allow for better group‐level assessments (such as the relative timing of biomarkers) rather than attributing variability to patient‐specific effects. Our approach was able to realign individuals on a common trajectory and amyloid‐centric timescale. Markov chain Monte Carlo sampling allows us to estimate uncertainties for all parameters: slopes and intercepts, changepoints, time‐shifts and the fit residual. Further work could explore the timing of other biomarker changes in this amyloid‐synchronised time.

Coarsening in zero-range processes

We prove a fluid limit describing coarsening for zero-range processes on a finite number of sites, with asymptotically constant jump rates. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is described by a piecewise linear trajectory in the simplex indexed by the sites. The linear coefficients are determined by the trace process of the underlying random walk on the subset of non-empty sites, and the trajectory reaches an absorbing configuration in finite time. A boundary of the simplex is called absorbing for the fluid limit if a trajectory started at a configuration in the boundary remains in it for all times. We identify the set of absorbing configurations and characterize the absorbing boundaries.

Extending growth mixture model to assess heterogeneity in joint development with piecewise linear trajectories in the framework of individual measurement occasions.

Almost always, developmental processes are multivariate in nature such that several outcomes and the development among these variables are correlated; therefore, empirical researchers often desire to examine two or more variables over time to understand how these outcomes and their change patterns are correlated. Multivariate growth models (MGMs) allow researchers to examine the correlations among developmental parameters. This study relaxes one population assumption of MGMs to investigate possible latent classes of joint development. The developed model enables the investigation of heterogeneity in the correlation between longitudinal outcomes and the effect of covariates on heterogeneity. More importantly, we propose using the piecewise linear functional form to estimate the stage-specific growth rates and stage-specific correlations. We demonstrate the proposed model through a simulation study and an analysis of real-world data. Our simulation study shows that the proposed model can separate joint development into multiple latent classes and provide unbiased and accurate point estimates with target coverage probabilities for the parameters of interest. Using longitudinal reading and mathematics scores from Grade K to 5, we demonstrate that the proposed model can capture heterogeneity in the correlation between joint development and estimate the stage-specific correlations. Additionally, we demonstrate how to identify the covariates that contribute the most to latent classes and transform candidate covariates from a large set to a manageable set while retaining the meaningful properties of the original covariate set for the mixture model with joint developmental processes. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Holistic Spatio-Temporal Graph Attention for Trajectory Prediction in Vehicle-Pedestrian Interactions.

Ensuring that intelligent vehicles do not cause fatal collisions remains a persistent challenge due to pedestrians' unpredictable movements and behavior. The potential for risky situations or collisions arising from even minor misunderstandings in vehicle-pedestrian interactions is a cause for great concern. Considerable research has been dedicated to the advancement of predictive models for pedestrian behavior through trajectory prediction, as well as the exploration of the intricate dynamics of vehicle-pedestrian interactions. However, it is important to note that these studies have certain limitations. In this paper, we propose a novel graph-based trajectory prediction model for vehicle-pedestrian interactions called Holistic Spatio-Temporal Graph Attention (HSTGA) to address these limitations. HSTGA first extracts vehicle-pedestrian interaction spatial features using a multi-layer perceptron (MLP) sub-network and max pooling. Then, the vehicle-pedestrian interaction features are aggregated with the spatial features of pedestrians and vehicles to be fed into the LSTM. The LSTM is modified to learn the vehicle-pedestrian interactions adaptively. Moreover, HSTGA models temporal interactions using an additional LSTM. Then, it models the spatial interactions among pedestrians and between pedestrians and vehicles using graph attention networks (GATs) to combine the hidden states of the LSTMs. We evaluate the performance of HSTGA on three different scenario datasets, including complex unsignalized roundabouts with no crosswalks and unsignalized intersections. The results show that HSTGA outperforms several state-of-the-art methods in predicting linear, curvilinear, and piece-wise linear trajectories of vehicles and pedestrians. Our approach provides a more comprehensive understanding of social interactions, enabling more accurate trajectory prediction for safe vehicle navigation.

Dynamic programming network for point target detection

To improve the efficiency of the dim point target detection based on dynamic programming (DP), this paper proposes a multi-frame target detection method based on a DP ring network (DPRN). In the proposed method, first, the target trajectory is approximated using the piecewise linear function. The velocity space partition DP (VSP-DP) is used to accumulate the merit functions of a target on each piecewise linear trajectory segment to avoid the merit function diffusion in different velocity spaces. In addition, the velocity space matching DP (VSM-DP) is employed to realize the state transition of a target between adjacent piecewise linear trajectory segments. Then, the VSP-DP and VSM-DP are used to construct a DP network (DPN). Second, to suppress the merit function diffusion further, the sequential and reverse DPNs are connected in a head-to-tail manner to form a DPRN, and the merit function of the DPRN is obtained by averaging the merit functions of the sequential and reverse DPNs. Finally, the target trajectory is obtained by tracking the extreme points of the merit functions of the DPRN. The simulation and analysis results show that the proposed DPRN combines the advantages of high detection probability of the high-order DP and high execution efficiency of the first-order DP. The proposed DPRN is suitable for radars and infrared searching and tracking systems.

Hierarchical Real-Time Optimal Planning of Collision-Free Trajectories of Collaborative Robots

In collaborative robotics the manipulator trajectory has to be planned to avoid collisions, yet in real-time. In this paper we pose the problem as minimization of a quadratic functional among piecewise linear trajectories in the angular (joint) space. The minimization is subjected to novel nonlinear inequality constraints that simplify the original non-penetration constraints to become cheap to evaluate in real time while still preserving collision-avoidance. The very first and most critical step of the computation is to find an initial trajectory that is free of collisions. To that goal we minimize a weighted sum of the violated constraints until they become feasible or a maximal number of steps is reached. Sometimes an incremental growing of the obstacle helps. By incremental growing we mean that we sequentially solve auxiliary subproblems with obstacles growing from ground or falling from top and use as the initial trajectory the one optimized in the previous step. The initial trajectory is then optimized while preserving feasibility at each step. We solve a sequence of simple-bound constrained quadratic programming problems formulated in the dual space of Lagrange multipliers, which are related to the original linearized inequality constraints that are active or close-to-active. Finally, we refine the trajectory parameterization and repeat the optimization, which we refer to as an hierarchical approach, until an overall prescribed time limit, being well below a second, is reached.

Algorithms for Planning Smoothed Individual Trajectories of Ground Robots

The article is devoted to the development of an algorithm for constructing the trajectory of a robotic platform moving in an environment with obstacles. This algorithm is based on the application of a special local optimization procedure at each planning step and allows us to obtain feasible program trajectories without increasing the computational complexity of algorithms compared to existing methods. The algorithm is based on the application of the improved method of potential fields and subsequent smoothing of the resulting trajectory. The improving of the potential field method consists in a new way of detecting and avoiding local minima. When a local minimum is detected, it is added to the map as an additional obstacle, which makes it possible to avoid it during further trajectory planning. To circumvent obstacles that can be approximated by polygons, the method of the effective point to the obstacle is proposed, which is the equivalent of the latter in relation to the current location of the moving robotic platform when using this planning method. A two-stage technique for smoothing piecewise linear trajectories is proposed. It is assumed that there is some initial suboptimal curve found by any planning method. This curve is optimized using a functional that includes the length of the trajectory and the deviation of the optimized curve from the original curve. At the second stage, the linear segments of the planned straight line are conjugated with second-order curves. As a result, the planned trajectory of motion is a quadratic-linear curve with a smooth function of the trajectory velocity. At the same time, the proposed method of coupling rectilinear sections of the trajectory does not require sudden changes in speed when passing turns. Simulation results confirming the effectiveness of the proposed method of planning the trajectories of robots are considered and discussed.

Supplemental Material for Extending Growth Mixture Model to Assess Heterogeneity in Joint Development With Piecewise Linear Trajectories in the Framework of Individual Measurement Occasions

Supplemental Material for Extending Growth Mixture Model to Assess Heterogeneity in Joint Development With Piecewise Linear Trajectories in the Framework of Individual Measurement Occasions

Affine body dynamics

Simulating stiff materials in applications where deformations are either not significant or else can safely be ignored is a fundamental task across fields. Rigid body modeling has thus long remained a critical tool and is, by far, the most popular simulation strategy currently employed for modeling stiff solids. At the same time, rigid body methods continue to pose a number of well known challenges and trade-offs including intersections, instabilities, inaccuracies, and/or slow performances that grow with contact-problem complexity. In this paper we revisit the stiff body problem and present ABD, a simple and highly effective affine body dynamics framework, which significantly improves state-of-the-art for simulating stiff-body dynamics. We trace the challenges in rigid-body methods to the necessity of linearizing piecewise-rigid trajectories and subsequent constraints. ABD instead relaxes the unnecessary (and unrealistic) constraint that each body's motion be exactly rigid with a stiff orthogonality potential, while preserving the rigid body model's key feature of a small coordinate representation. In doing so ABD replaces piecewise linearization with piecewise linear trajectories. This, in turn, combines the best of both worlds: compact coordinates ensure small, sparse system solves, while piecewise-linear trajectories enable efficient and accurate constraint (contact and joint) evaluations. Beginning with this simple foundation, ABD preserves all guarantees of the underlying IPC model we build it upon, e.g., solution convergence, guaranteed non-intersection, and accurate frictional contact. Over a wide range and scale of simulation problems we demonstrate that ABD brings orders of magnitude performance gains (two- to three-orders on the CPU and an order more when utilizing the GPU, obtaining 10, 000× speedups) over prior IPC-based methods, while maintaining simulation quality and nonintersection of trajectories. At the same time ABD has comparable or faster timings when compared to state-of-the-art rigid body libraries optimized for performance without guarantees, and successfully and efficiently solves challenging simulation problems where both classes of prior rigid body simulation methods fail altogether.

Prerequisites to Design a Collision Free Trajectory in a 3D Dynamic Environment for an UAV

Abstract This research presents the main steps needed for designing a piece-wise linear trajectory which grants an unmanned aerial vehicle (UAV) to reach a destination pose in a workspace with dynamic obstacles. The first objective is characterized by the examination of kinematics and dynamics of a quadcopter. For this purpose, a nonlinear mathematical model was developed. The validation of the mathematical model was confirmed by MATLAB and real time experiments. The second step aims to properly map the 3D environment. For this objective, an algorithm for a cuboid rectangular decomposition was developed and implemented, by extending a 2D decomposition technique. The evaluation of the proposed path planning algorithm occurred for different scenarios and the validation of the results was established through numerical simulations. In the end, a comparison for two path planning scenarios is shown: for a 3D static environment and for a 3D dynamic environment, where the movement of a dynamic obstacle is known.

Diffusion SLAM: Localizing Diffusion Sources From Samples Taken by Location-Unaware Mobile Sensors

We consider diffusion fields induced by multiple localised and instantaneous sources. We assume a mobile sensor samples the field, uniformly along a piecewise linear trajectory, which is unknown. The problem we address is the estimation of the amplitudes and locations of the diffusion sources, as well as of the trajectory of the sensor. We first propose a method for diffusion source localisation and trajectory mapping (D-SLAM) in 2D, where we assume the activation times of the sources are known and the evolution of the diffusion field over time is negligible. The reconstruction method we propose maps the measurements obtained using the mobile sensor to a sequence of generalised field samples. From these generalised samples, we can then retrieve the locations of the sources as well as the trajectory of the sensor (up to a 2D orthogonal geometric transformation). We then relax these assumptions and show that we can perform D-SLAM also in the case of unknown activation times, from samples of a time-varying field, as well as in 3D spaces. Finally, simulation results on both synthetic and real data further validate the proposed framework.

Pareto Optimization of Cycle Time and Motion Accuracy in Trajectory Planning for Industrial Feed Drive Systems

Manufacturing industries aim to improve product quantity and quality. These trade-off objectives are typically manifested as cycle time reduction and motion accuracy improvement. Corner smoothing approaches that reduce the cycle time of piecewise linear trajectories are proposed in the literature. This study tackles the two objectives by Pareto-optimal corner smoothing with constraints imposed as kinematic limits, continuity conditions and user-specified cornering tolerance. Linear and cornering motions along a contour are respectively described by jerk-limited acceleration profiles and a modified kinematic corner smoothing with interrupted acceleration (KCSIA) approach. A Pareto frontier is generated by the divide and conquer algorithm, where the solution nearest to the utopia point is selected as the best trade-off solution. The effectiveness of the proposed method is validated through experiments, where the best trade-off solution reduces the maximum and average contouring errors by 47.53% and 25.40% while it increases cycle time by 2.53% compared to KCSIA.

Human Perception-Optimized Planning for Comfortable VR-Based Telepresence

This paper introduces an emerging motion planning problem by considering a human that is immersed into the viewing perspective of a remote robot. The challenge is to make the experience both effective (such as delivering a sense of presence) and comfortable (such as avoiding adverse sickness symptoms, including nausea). We refer to this challenging new area as human perception-optimized planning and propose a general multiobjective optimization framework that can be instantiated in many envisioned scenarios. We then consider a specific VR telepresence task as a case of human perception-optimized planning, in which we simulate a robot that sends 360 video to a remote user to be viewed through a head-mounted display. In this particular task, we plan trajectories that minimize VR sickness (and thereby maximize comfort). An A* type method is used to create a Pareto-optimal collection of piecewise linear trajectories while taking into account criteria that improve comfort. We conducted a study with human subjects touring a virtual museum, in which paths computed by our algorithm are compared against a reference RRT-based trajectory. Generally, users suffered less from VR sickness and preferred the paths created by the presented algorithm.

A Novel State Transformation Approach to Tracking of Piecewise Linear Trajectories

In this paper, we propose a novel approach for tracking of piecewise linear trajectories, such as triangular and staircase waveforms. We derive state and input transformations, which result in closed-loop error dynamics driven by a series of impulses. The proposed control structure takes the form of an output-feedback-feedforward system that is straightforward to implement. In contrast to the recently proposed tracking control methods for such trajectories, the closed-loop stability is not affected by the frequency of the desired triangular reference. The method is implemented on a nanopositioner serving as the scanning stage of an atomic force microscope.

Hold Or take Optimal Plan (HOOP): A quadratic programming approach to multi-robot trajectory generation

In this work, we present Hold Or take Optimal Plan (HOOP), a centralized trajectory generation algorithm for labeled multi-robot systems operating in obstacle-free, two-dimensional, continuous workspaces. Given a team of N robots, each with nth-order dynamics, our algorithm finds trajectories that navigate vehicles from their start positions to non-interchangeable goal positions in a collision-free manner. The algorithm operates in two phases. In the motion planning step, a geometric algorithm finds a collision-free, piecewise-linear trajectory for each robot. In the trajectory generation step, each robot’s trajectory is refined into a higher-order piecewise polynomial with a quadratic program. The novelty of our method is in this problem decomposition. The motion plan, through abstracting away robots’ dynamics, can be found quickly. It is then subsequently leveraged to construct collision avoidance constraints for N decoupled quadratic programs instead of a single, coupled optimization problem, decreasing computation time. We prove that this method is safe, complete, and generates smooth trajectories that respect robots’ dynamics. We demonstrate the algorithm’s practicality through extensive quadrotor experiments.

Transition state clustering: Unsupervised surgical trajectory segmentation for robot learning

Demonstration trajectories collected from a supervisor in teleoperation are widely used for robot learning, and temporally segmenting the trajectories into shorter, less-variable segments can improve the efficiency and reliability of learning algorithms. Trajectory segmentation algorithms can be sensitive to noise, spurious motions, and temporal variation. We present a new unsupervised segmentation algorithm, transition state clustering (TSC), which leverages repeated demonstrations of a task by clustering segment endpoints across demonstrations. TSC complements any motion-based segmentation algorithm by identifying candidate transitions, clustering them by kinematic similarity, and then correlating the kinematic clusters with available sensory and temporal features. TSC uses a hierarchical Dirichlet process Gaussian mixture model to avoid selecting the number of segments a priori. We present simulated results to suggest that TSC significantly reduces the number of false-positive segments in dynamical systems observed with noise as compared with seven probabilistic and non-probabilistic segmentation algorithms. We additionally compare algorithms that use piecewise linear segment models, and find that TSC recovers segments of a generated piecewise linear trajectory with greater accuracy in the presence of process and observation noise. At the maximum noise level, TSC recovers the ground truth 49% more accurately than alternatives. Furthermore, TSC runs 100× faster than the next most accurate alternative autoregressive models, which require expensive Markov chain Monte Carlo (MCMC)-based inference. We also evaluated TSC on 67 recordings of surgical needle passing and suturing. We supplemented the kinematic recordings with manually annotated visual features that denote grasp and penetration conditions. On this dataset, TSC finds 83% of needle passing transitions and 73% of the suturing transitions annotated by human experts.

Approach to detector design for statistical multiple‐input–multiple‐output radars using multi‐scan data

This study considers the detector design and performance analysis in a statistical multiple-input-multiple-output radar. The authors studied this subject under some assumptions such as single point target, Swerling scattering models for radar cross-section and Gaussian interference. The target is assumed to move in a piecewise linear trajectory and multi-scan data is stored. So, the target cell may have changed during the time. A three-dimensional data signal space is generated that involved in observations from multiple scans, then a generalised likelihood ratio test detector is derived that employs consecutive observations in a central unit. Then a sub-optimum detector is introduced in which each pair of transmitters-receivers works separately and the central unit decides with a binary algorithm. The exact theoretical performances of both detectors are extracted and their qualities are compared. Numerical simulations verify all results.

Real-time repairable interpolation scheme for CNC tool path processing

G01 trajectory is widely-adopted for CNC machining. The piecewise linear trajectory is geometrically discontinuous, bringing about drastic fluctuation of feedrate and acceleration, and causing low machining efficiency and quality. Many methods of trajectory smoothing are developed and applied in CNC system. However some shortcomings exist when these methods were applied in CNC machining. In the existing methods, the following requirements are not satisfied simultaneously: not less than G2 continuity, confined chord error, shape preservation, interpolating G01 points, spatial applicability, real-time performance of algorithm, free geometric form. Considering various requirements of CNC machining, this work develops an interpolation scheme with local repairing function to deal with the possible shape deficiencies of trajectory. The resulting tool path has G2 continuity and is optimized in bending energy under shape-preserving and chord error constraints. Combining with jerk-limited feedrate scheduling, a real-time tool path processing strategy is developed. The simulation demonstrates the proposed method eliminates shape deficiencies, and has apparent advantage in the reduction of contour error, curvature, bending energy and machining time, compared with Zhao’s and Fan’s methods. The experiment demonstrates the advantages of the proposed algorithm in improving the quality of machined surface and machining efficiency, compared with previous methods.

A two-stage model in a Bayesian framework to estimate a survival endpoint in the presence of confounding by indication.

Background Biomarker series can indicate disease progression and predict clinical endpoints. When a treatment is prescribed depending on the biomarker, confounding by indication might be introduced if the treatment modifies the marker profile and risk of failure. Objective Our aim was to highlight the flexibility of a two-stage model fitted within a Bayesian Markov Chain Monte Carlo framework. For this purpose, we monitored the prostate-specific antigens in prostate cancer patients treated with external beam radiation therapy. In the presence of rising prostate-specific antigens after external beam radiation therapy, salvage hormone therapy can be prescribed to reduce both the prostate-specific antigens concentration and the risk of clinical failure, an illustration of confounding by indication. We focused on the assessment of the prognostic value of hormone therapy and prostate-specific antigens trajectory on the risk of failure. Methods We used a two-stage model within a Bayesian framework to assess the role of the prostate-specific antigens profile on clinical failure while accounting for a secondary treatment prescribed by indication. We modeled prostate-specific antigens using a hierarchical piecewise linear trajectory with a random changepoint. Residual prostate-specific antigens variability was expressed as a function of prostate-specific antigens concentration. Covariates in the survival model included hormone therapy, baseline characteristics, and individual predictions of the prostate-specific antigens nadir and timing and prostate-specific antigens slopes before and after the nadir as provided by the longitudinal process. Results We showed positive associations between an increased prostate-specific antigens nadir, an earlier changepoint and a steeper post-nadir slope with an increased risk of failure. Importantly, we highlighted a significant benefit of hormone therapy, an effect that was not observed when the prostate-specific antigens trajectory was not accounted for in the survival model. Conclusion Our modeling strategy was particularly flexible and accounted for multiple complex features of longitudinal and survival data, including the presence of a random changepoint and a time-dependent covariate.

Numerical Inversion of a Broken Ray Transform Arising in Single Scattering Optical Tomography

The article presents an efficient image reconstruction algorithm for single scattering optical tomography (SSOT) in circular geometry of data acquisition. This novel medical imaging modality uses photons of light that scatter once in the body to recover its interior features. The mathematical model of SSOT is based on the broken ray (or V-line Radon) transform (BRT), which puts into correspondence to an image function its integrals along V-shaped piecewise linear trajectories. The process of image reconstruction in SSOT requires inversion of that transform. We implement numerical inversion of a broken ray transform in a disc with partial radial data. Our method is based on a relation between the Fourier coefficients of the image function and those of its BRT recently discovered by Ambartsoumian and Moon. The numerical algorithm requires solution of ill-conditioned matrix problems, which is accomplished using a half-rank truncated singular value decomposition method. Several numerical computations validating the inversion formula are presented, which demonstrate the accuracy, speed, and robustness of our method in the case of both noise-free and noisy data.