Non-smooth Trajectories Research Articles

Trajectory optimization of B-splines interpolation based on dynamic error adjustment

In this paper, we propose a motion planning algorithm based on the five-fold B-spline curve for multi-axis motion control systems with non-smooth trajectories, acceleration, plus acceleration over the limit, etc. We construct a smooth transition model for the micro-line segment G3 of the continuous five-fold B-spline curve. A derivation of the corresponding theoretical equation is given. Using the transition model, we propose an optimization method for dynamic adjustment of motion trajectory errors based on the velocity forward planning constraints by smoothing the transition after the trajectory of the micro-line segment for the forward planning velocity. Finally, the design flow for the forward planning of the velocity is given. Finally, the above algorithm is used to perform simulation experiments for the optimization algorithm. Experimental results show that the proposed optimization improves the accuracy of fitting motion trajectories under the constraints of acceleration and deceleration addition, which validates the proposed algorithm.

Non-Smooth Trajectory Optimization for Wheeled Balancing Robots with Contact Switches and Impacts

Non-Smooth Trajectory Optimization for Wheeled Balancing Robots with Contact Switches and Impacts

Dynamic Models with Sigmoid Corrections to Generation of an Achievable 4D-Trajectory for a UAV and Estimating Wind Disturbances

For an unmanned aerial vehicle (UAV) of an aircraft type, the problems of planning achievable trajectories as well as robust control under wind disturbances are considered. A computationally simple method for compiling a primary non-smooth 4D trajectory is proposed. Its segments connect the given waypoints and determine the desired average velocity in various sections. Instead of time-consuming methods of analytical smoothing of broken path joints using polynomials, a tracking differentiator with S-shaped smooth and limited sigmoid corrective actions is developed. This virtual dynamic model provides natural smoothing of the primary trajectory considering the design constraints on the velocity, acceleration, and thrust of the UAV. The tracking differentiator variables create an achievable trajectory and are used to synthesize the UAV tracking system. To compensate for the action of wind disturbances on the UAV, a disturbance observer is developed. It is a replica of the equations of the control plant model, which are directly affected by external uncontrolled disturbances. These algorithms also use sigmoid corrections. Unlike standard disturbances observers, this approach does not require the development of a dynamic model of external disturbances and does not assume their smoothness. The effectiveness of the developed algorithms was confirmed by numerical simulation.

Feature-Based Direct Tracking and Mapping for Real-Time Noise-Robust Outdoor 3D Reconstruction Using Quadcopters

In this work, we focus on real-time 3D reconstruction or localization and mapping for outdoor scene using an aerial vehicle called quadcopter. Quadcopter provides the advantages of high flexibility and wide view field in spatial movement. However, existing feature-based and direct methods (using dense or semi-dense approach) are not suitable for outdoor environment, in which multiple challenging scenarios arise such as lighting variance, jittering views, high-speed and non-smooth flight trajectory. The main reason is that the existing methods rely on the assumption of brightness constancy across multiple images and only raw pixel intensities are employed for direct image alignment. In order to tackle these scenarios, a novel method called Feature-based Direct Tracking and Mapping (FDTAM) is proposed, which i) incorporates an efficient binary feature descriptor into direct image alignment module to tackle the challenging scenarios, such as drifting issue under lighting variance problem; ii) applies semi-dense approach to obtain high reconstruction quality; iii) provides a framework with low computational complexity for real-time reconstruction. Compared to other state-of-the-art feature-based and direct methods, our proposed method is shown to tackle the challenging scenarios and improve the accuracy and robustness even in CPU (rather than GPU) platform.

The structure of abnormal extremals in a sub-Riemannian problem with growth vector

A left-invariant sub-Riemannian problem on a free nilpotent Lie group of step 4 with two generators is considered. The structure of abnormal extremals is described. These extremals are shown to define an abnormal foliation of the annihilator of the square of the distribution, which is given by the intersections of this annihilator with the leaves of the symplectic foliation of the Lie coalgebra. The question of abnormal trajectories being strictly/nonstrictly abnormal is investigated, their projection onto a plane of the distribution are described, estimates for the corank are given and examples of nonsmooth trajectories are constructed. Bibliography: 14 titles.

Suppressing homoclinic chaos for a weak periodically excited non-smooth oscillator

In this work, some new effective methods for suppressing homoclinic chaos in a weak periodically excited non-smooth oscillator are studied, and the main idea is to modify slightly the Melnikov function such that the zeros are eliminated. Firstly, a general form of planar piecewise-smooth oscillators is given to approximatively model many nonlinear restoring force of smooth oscillators subjected to all kinds of damping and periodic excitations. In the absence of controls, the Melnikov method for non-smooth homoclinic trajectories within the framework of a piecewise-smooth oscillator is briefly introduced without detailed derivation. This analytical tool is useful to detect the threshold of parameters for the existence of homoclinic chaos in the non-smooth oscillator. After some methods of state feedback control, self-adaptive control and parametric excitations control are, respectively, considered, sufficient criteria for suppressing homoclinic chaos are derived by employing the Melnikov function of non-smooth systems. Finally, the effectiveness of strategies for suppressing homoclinic chaos is analytically and numerically demonstrated through a specific example.

Iterative NC program modification and energy saving for a CNC machine tool feed drive system with linear motors

This study proposes a design of iterative learning contouring controller (ILCC) by considering actual contour error compensation (ACEC) to enhance the contouring performance of CNC machine tool feed drive systems with linear motors. The ACEC with linear and circular interpolation is designed to estimates contour error precisely. The proposed control iteratively modifies the numerical control (NC) programs for each drive axis to reduce a contour error. Hence, the proposed approach can be directly applied for a commercial CNC machine tool with linear motors currently in use without any modification of their original controllers. Both of linear and circular interpolations are verified by simulation in both air-cutting and machining condition. The simulation is conducted for a non-smooth rhombus and circular trajectory. The effectiveness of the proposed methods has been experimentally verified through a CNC machine tool with linear motors for a non-smooth rhombus trajectory. Experimental results show that the proposed controller could reduce the maximum and mean contour errors by 94.58% and 88.67% on average, respectively. The proposed method improved the control input variance by 37.9%, and consequently energy consumption was reduced by 11.7% compared with the original NC program.

Switching Regimes in Economics: The Contraction Mapping and the ω-Limit Set

This paper considers a dynamical system defined by a set of ordinary autonomous differential equations with discontinuous right-hand side. Such systems typically appear in economic modelling where there are two or more regimes with a switching between them. Switching between regimes may be a consequence of market forces or deliberately forced in form of policy implementation. Stiefenhofer and Giesl [1] introduce such a model. The purpose of this paper is to show that a metric function defined between two adjacent trajectories contracts in forward time leading to exponentially asymptotically stability of (non)smooth periodic orbits. Hence, we define a local contraction function and distribute it over the smooth and nonsmooth parts of the periodic orbits. The paper shows exponentially asymptotical stability of a periodic orbit using a contraction property of the distance function between two adjacent nonsmooth trajectories over the entire periodic orbit. Moreover it is shown that the ω-limit set of the (non)smooth periodic orbit for two adjacent initial conditions is the same.

A new diamond machining approach for extendable fabrication of micro-freeform lens array

Abstract Fast tool servo diamond turning is conventionally a spiral path-based process widely used to efficiently fabricate non-rotationally symmetric surfaces with ultra-fine accuracy. However, the technique is naturally restricted by the demanding bandwidth and resolution requirements, which are directly related to the complexity and size of the machining surface. Moreover, non-smooth trajectories required for machining complex lens array with discontinuities, due to overlapping sharp edges, induce vibrations causing the surface quality to deteriorate, and adding additional barriers to using this conventional technique in broader applications. In this paper, a new diamond machining approach is proposed to address its inherent drawbacks. Using the new method, complex lens array consisting of arbitrary lenslet shapes could be fabricated in an extendable way without the typical limitations. Theoretical investigations of the unique cutting kinematics, motion determination, servo synchronization and compensation will be carried out. To predict the machined surface topography, the surface generation mechanism with consideration of both the kinematic tool/workpiece interaction and elastic recovery of materials is presented. The experimental fabrication of hexagonal micro-sphere lens array, micro-freeform lens array, as well as seamlessly stitched sinusoidal surface with quadrupled areas was highly consistent and aligned with the theoretical predictions, thus demonstrating the effectiveness of the new approach and its potential for broader applications.

Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on Lie group SE(2)

ABSTRACTThis paper studies the problem of set-point stabilisation, trajectory tracking, and formation tracking for unicycle-type vehicles by taking advantage of the exponential coordinates and the other mathematical tools provided by the Lie group setting of the special Euclidean group SE(2). Motivated by recent developments in geometric control approaches for holonomic systems, we first study the stabilisation problem for nonholonomic constrained kinematic systems by improving the logarithmic stabilising control laws in such a way that they can satisfy nonholonomic constraints too. We then extend the control design to the problem of trajectory tracking by proposing auxiliary systems and investigating the conditions for which the adjoint map in the Lie algebra preserves the velocity constraint. This leads to a global control law which is valid for general time-varying as well as non-smooth trajectories. The tracking control law is also applied to the problem of leader-follower formation tracking by constructing nonholonomic virtual leaders in the desired formation. Finally, the kinematic control law is translated to a differential drive robot using the backstepping technique. To demonstrate the effectiveness of the controllers, the numerical results are presented and the performances are compared with the other three controllers in the literature.

Tracking by switching state space models

We propose a novel tracking method that allows to switch between different state representations as, e.g., image coordinates in different views or image and ground plane coordinates. During the tracking process, our method adaptively switches between these representations. We demonstrate the applicability of our method for dynamic cameras tracking dynamic objects: Using the image based representation (non-smooth trajectories if the camera is shaking) together with the ground plane based one (estimation uncertainty in visual odometry or ground plane orientation), the disadvantages of both representation forms can be overcome: Non-occluded observations on the image plane provide strong appearance cues for the target. Smooth paths on the ground plane provide strong motion cues with the camera motion factored out. Following a Bayesian tracking approach, we propose a probabilistic framework that determines the most appropriate state space model (SSM)—image or ground plane or both—at each time instance. Experimental results demonstrate that our method outperforms the state-of-the-art.

PSO-Lyapunov motion/force control of robot arms with model uncertainties

SUMMARYA method for motion/force control of robot arms with model uncertainties is presented. Tracking control of complex trajectories is guaranteed using a Lyapunov approach with high-precision performance ensured using a particle swarm optimization (PSO) algorithm. Tracking performance and robustness are simulated for a robotic device for limb rehabilitation that is designed to be adapted easily to different subjects by considering model parameter uncertainties. Controller parameters are optimized offline using the PSO algorithm with Lyapunov stability conditions considered as inequality constraints. Using the control scheme, the robot can guide limbs on smooth and non-smooth trajectories, under model uncertainties and measurement noise.

Control of Redundant Joint Structures Using Image Information During the Tracking of Non-Smooth Trajectories

Visual information is increasingly being used in a great number of applications in order to perform the guidance of joint structures. This paper proposes an image-based controller which allows the joint structure guidance when its number of degrees of freedom is greater than the required for the developed task. In this case, the controller solves the redundancy combining two different tasks: the primary task allows the correct guidance using image information, and the secondary task determines the most adequate joint structure posture solving the possible joint redundancy regarding the performed task in the image space. The method proposed to guide the joint structure also employs a smoothing Kalman filter not only to determine the moment when abrupt changes occur in the tracked trajectory, but also to estimate and compensate these changes using the proposed filter. Furthermore, a direct visual control approach is proposed which integrates the visual information provided by this smoothing Kalman filter. This last aspect permits the correct tracking when noisy measurements are obtained. All the contributions are integrated in an application which requires the tracking of the faces of Asperger children.

Customized Fetal Growth Modeling and Monitoring—A Statistical Process Control Approach

ABSTRACT A statistical process control (SPC)-based methodology is developed to detect early deviations of fetal biometry from expected normal growth. Using response modeling methodology (RMM), a fetal growth model is dynamically estimated and integrated in a regression-adjusted SPC control scheme, based on a new median control chart and a control chart for residuals variation. Hadlock's reference centiles are also integrated in the monitoring scheme. Longitudinal data from normal pregnancies and those with adverse medical outcomes have been analyzed. Results show that nonsmooth growth trajectory, expressed in exceptionally large absolute deviations from predicted median values, is a good precursor to possible prenatal adverse outcomes.

Monitoring structural deterioration of railway turnout systems via dynamic wheel/rail interaction

‘Big data’ obtained from wayside detection systems and sensors installed on board a train show that actual loading history for a railway track is rather dynamic and transient. The dynamic loadings due to train and track interactions redistribute from the rails to the rail pad, from the rail pad to the railway sleeper, and from the railway sleeper to the underlying ground. Dynamic content redistributed onto each layer of track is also filtered by energy dissipation characteristic of materials and structures. As a critical infrastructure, railway turnout is a structural grillage system used to divert a train to other directions or other tracks. The wheel–rail contact over the crossing transfer zone often causes detrimental impact loads that rapidly deteriorate the turnout and its components. The dynamic responses of wheel–rail interaction depend largely on the non-smooth trajectory or wearing condition of crossing geometry. In reality, a railway line spreads over a large distance and monitoring such rail infrastructure is one of the challenges in rail industry. This paper presents a methodology and application to evaluate and monitor the structural deterioration of railway turnout systems in an Australian urban rail network. The method has integrated numerical train/track simulations, axle box acceleration and ride quality data obtained from the calibrated track inspection vehicle “AK Car”.

A boundary integral formulation for particle trajectories in Stokes waves

Recently important theorems have been established presenting qualitative results for particle trajectories below a Stokes wave. A diversity of orbit patterns were described, including the case of a closed orbit when a Stokes wave propagates in the presence of an adverse current. In this work these results are revisited in a quantitative fashion through a boundary integral formulation which leads to very accurate numerical simulations of particle trajectories. The boundary integral formulation allows the accurate evaluation of the vector field of the (particle's) dynamical system, without resorting to a series expansion and a small parameter. Accurate trajectories are benchmarked against well known expansions for weakly nonlinear waves. Simulations are then performed beyond this regime. Closed orbits are found in the presence of an adverse current, as well as non-smooth trajectories that have not been reported. These occur for both adverse and favorable currents.

Smooth Trajectory Planning for 3-UPU Parallel Manipulator

Exciting the mechanical resonance mode and vibration caused by non-smooth trajectories can cause some failure in the structure of the robot and its actuators. This work proposes a smooth trajectory planning for 3 degree of freedom (dof) 3-upu parallel manipulator. We have used optimization method to define the smooth trajectory for this robot. In this paper an objective function has been used which contain a term related to the total execution time and a term corresponding to the integral of squared jerk. Some constraints have been applied to the problem as the input of our system. Task space and joint space trajectory are then considered as the output. In this method optimization techniques such as genetic algorithm and fmincon (function of MatLabTM ) have been investigated. Six via points have been considered and B-splines have been used to present kinematic quantities.

Tool path optimization in postprocessor of five-axis machine tools

Generally, tool path is generated in a computer-aided manufacturing software considering only the geometry of machining parts. It is converted into numerical control (NC) codes in the postprocessor based on the particular machine kinematics. For some special types of five-axis machine tools, e.g., non-orthogonal five-axis machine tools, the generated NC codes may produce unqualified parts because of the existence of the non-linear error. Conventional commercialized postprocessors usually do not have the function of non-linear error checking. Observing that the tool path is a non-smooth trajectory full of corners and a series of connected line segments, cubic spline interpolation is applied to smooth the tool path at regular points in this study. The cutter tip center points are computed by the cubic spine interpolation, while the cutter posture vectors are obtained via linear interpolation. At the splines (for regular points) and the line segments (feature points), more points are chosen to be converted into NC codes to reduce the non-linear error, which is called data densification. Using the cubic spline to smooth the tool path and the data densification to reduce the non-linear error, a novel tool path optimization algorithm in postprocessor is proposed. Experiments were carried out on an inclined rotary spindle axis non-orthogonal five-axis machine tool. It shows that the proposed tool path optimization provides improved accuracy and surface quality.

Controller design for two-wheeled self-balancing vehicles using feedback linearisation technique

Two-wheeled self-balancing vehicles have been extensively used because of their unique capabilities such as reducing the traffic problems and being environment friendly. The main challenge in these vehicles is to design robust controllers capable of creating smooth and safe movement. This paper focuses on design of such a controller using the feedback linearisation technique. The modelled system comprises of a single-link inverted pendulum, which simulates the rider’s body, mounted atop a two-wheeled platform. While the base is being externally disturbed, the designed controllers guarantee the stability of the system and keep the rider along its equilibrium situation. In order to validate the effectiveness of proposed control schemes, some sets of simulation studies are carried out on different smooth and non-smooth surfaces. It is finally shown that besides having stability on smooth surfaces, the system behaves stably on non-smooth trajectories.

Smooth time-optimal tool trajectory generation for CNC manufacturing systems

An optimization approach is proposed in this paper for generating smooth and time-optimal path constrained tool trajectory for Cartesian computer numerical control (CNC) manufacturing systems. The desired smooth time-optimal trajectory generation (STOTG) problem is formulated as a general optimal control problem. And axis jerk (derivative of acceleration with respect to time) constraints are introduced into this problem to remove discontinuities of the acceleration profiles. The desired smoothness of the trajectory can be accomplished by adjusting the values of jerk constraints. A control vector parameterization (CVP) method is applied to convert the optimal control problem into a nonlinear programming (NLP) problem which can be solved conveniently and effectively. The third derivative of the path parameter with respect to time (pseudo-jerk) and jerk act as optimization variables. The pseudo-jerk is approximated as piecewise constant, thus for at least second-order continuous parametric path, the resulted optimized trajectory with respect to time is also at least second-order continuous. Sequential quadratic programming (SQP) method is used to solve the NLP problem, through which numerical solution is obtained. Non-smooth (i.e. without considering jerk constraints) time-optimal trajectory generation (non-STOTG) problem is also considered in this paper for the purpose of comparison. Solutions of time-optimal trajectory generation (TOTG) problems for two test paths are performed to verify the effectiveness of the proposed approach.