A Geometric Approach for Error Space Estimation of Planar Linkage

This article develops a geometric method to estimate the clearances-induced error space of any planar linkage. The error space discussed here represents the unconstrained mobility of the end-effector when actuators of the mechanism are locked, and is expressed by a connected geometry in 3-dimensional Euclidean frame {x, y, θ}. First, error space of the planar mechanism is modeled and closed-form expressions are derived. Then, levels of joints in error propagation analysis are defined and illustrated with an example of a eight-bar linkage, following which error propagation path among closed-loop structures is given. The modeling of error propagation and accumulation is introduced in detail. Moreover, a simplification technique is discussed for simple expression of the error space propagated from previous joints. This study provides a way to have a deep insight into the accuracy performance of any planar linkage and the proposed error space evaluation method is validated by case study of error space estimation of a four-bar linkage and a six-bar linkage. For the four-bar linkage, the structure with optimal accuracy is obtained. And for the six-bar linkage, the error space of the end-effector is expressed in closed form and visualized in the 3-dimensional frame. Finally, this work is concluded and advances of the proposed method are emphasized.

ANALISIS KESALAHAN FAKTA DAN KESALAHAN KONSEP MAHASISWA DALAM MENYELESAIKAN SOAL GEOMETRI ANALITIK RUANG

This study aims to describe both fact and concept error of students in solving space analytic geometry problems. The subjects of this research are two students of Mathematics Department of Universitas Negeri Makassar. Each of them represents for each error (fact and concept errors). The collecting data were employed by using space analytic geometric tests and depth-interview. Interview guidelines and researcher were as research instruments. Data were qualitatively analyzed, using three stages of analysis: data reduction, data display and concluding. The main findings of this research are (1) the subject of fact error made mistakes in writing vector symbols, (2) the subjects of concept error made mistakes in identifying an equation (plane equation or line equation), since his focus was only in the number of variables of the equation.Keywords: fact error, concept error, space analytic geometry

Dimensionality Reduction of Spectral Reflectance by Dividing the Error Space of Principal Component Analysis

A method based on the error space division of principal component analysis (PCA) is proposed to improve both the spectral and colorimetric reconstruction accuracy in spectral dimensionality reduction. Founded on the error source analysis of PCA from a geometric point of view, an objective function minimizing the within-cluster spectral reconstruction error is established to divide the error space of PCA. PCA is implemented again to each divided cluster to reduce the dimensionality of spectral reflectance. The proposed method, error-space-divided PCA (ESDPCA), is tested using four different spectral datasets. The root mean squared error (RMSE) and CIEDE2000 colour difference are adopted as the spectral and colorimetric evaluation metric respectively. Statistical results indicate that ESDPCA can outperform PCA by at least one principal component (PC) in colorimetric accuracy, while it can outperform PCA by at least two or three PCs in spectral accuracy. Comparisons with other three representative methods (i.e., LabPQR, LabRGB, and XYZLMS) show that ESDPCA outperforms them both in spectral and colorimetric accuracy significantly. In addition, the proposed method is robust for spectral datasets and compatible with few other methods involving PCA. Moreover, the computation complexity of ESPCA has the same order of magnitude as that of PCA.

Robust coding with spiking networks: a geometric perspective

The interactions of large groups of spiking neurons have been difficult to understand or visualise. Using simple geometric pictures, we here illustrate the spike-by-spike dynamics of networks based on efficient spike coding, and we highlight the conditions under which they can preserve their function against various perturbations. We show that their dynamics are confined to a small geometric object, a ‘convex polytope’, in an abstract error space. Changes in network parameters (such as number of neurons, dimensionality of the inputs, firing thresholds, synaptic weights, or transmission delays) can all be understood as deformations of this polytope. Using these insights, we show that the core functionality of these network models, just like their biological counterparts, is preserved as long as perturbations do not destroy the shape of the geometric object. We suggest that this single principle—efficient spike coding—may be key to understanding the robustness of neural systems at the circuit level.

Dimensional Optimization for Minimally Invasive Surgery Robot Based on Double Space and Kinematic Accuracy Reliability Index

To improve the kinematic performance of the remote center mechanism for surgical robot, a double space index and kinematic accuracy reliability index are proposed to optimize the dimensional sizes of mechanism. First, the influence of the angular error on the position error and the operability of the remote center in the workspace are analyzed. The position error space and operability space index are weighted to establish the double space index. Second, a kinematic accuracy reliability index is established based on the influence of joint clearance on output position accuracy. Finally, the dimensional sizes of remote center adjustment mechanism and double parallelogram mechanism are optimized based on proposed optimization indices. Multipopulation genetic algorithm is used to obtain the optimal size parameters under the corresponding index. The optimized double space index is 56.7%, which is 56.5% higher than before optimization. The optimized kinematic accuracy reliability is 0.91, which is 22.9% higher than before optimization. The kinematic performance of remote center mechanism has been significantly improved after optimization.

Leader-Following Consensus and Formation Control of VTOL-UAVs with Event-Triggered Communications

This article presents the design and implementation of an event-triggered control approach, applied to the leader-following consensus and formation of a group of autonomous micro-aircraft with capabilities of vertical take-off and landing (VTOL-UAVs). The control strategy is based on an inner–outer loop control approach. The inner control law stabilizes the attitude and position of one agent, whereas the outer control follows a virtual leader to achieve position consensus cooperatively through an event-triggered policy. The communication topology uses undirected and connected graphs. With such an event-triggered control, the closed-loop trajectories converge to a compact sphere, centered in the origin of the error space. Furthermore, the minimal inter-sampling time is proven to be below bounded avoiding the Zeno behavior. The formation problem addresses the group of agents to fly in a given shape configuration. The simulation and experimental results highlight the performance of the proposed control strategy.