Dynamic modeling of competing technology designs, pricing and consumer dynamics

Geometric ModelingGeometric modeling is a crucial aspect of computer graphics and design, playing a pivotal role in various industries such as architecture, engineering, and entertainment. It involves the creation and manipulation of digital representations of geometric shapes and structures, allowing for the visualization and analysis of complex concepts. The significance of geometric modeling is evident in its widespread application, from the development of 3D models for video games to the design of intricate architectural structures. This article aims to explore the multifaceted nature of geometric modeling, delving into its technical, artistic, and practical implications. From a technical standpoint, geometric modeling encompasses a diverse range of methodologies and algorithms thatfacilitate the generation of digital representations of physical objects. These techniques include constructive solid geometry (CSG), boundary representation (B-rep), and parametric modeling, each offering distinct advantages in terms of precision, flexibility, and computational efficiency. The utilization of these methods is integral to the accurate portrayal of complex geometries, ensuring that digital models faithfully reflect their real-world counterparts. Moreover, advancements in geometric modeling have led to the development of sophisticated software tools such as computer-aided design (CAD) programs, enabling engineers and designers to create intricate models with unparalleled detail and accuracy. Beyond its technical intricacies, geometric modeling also encompasses an artistic dimension, serving as a medium for creative expression and aesthetic exploration. In the realm of digital art and animation, geometric modeling provides artists with the means to sculpt and manipulate virtual forms, breathing life into imaginary worlds and characters. The fusion of technical precision and artistic ingenuity is exemplified in the creation of visually stunning 3D models for animated films and video games, where geometric modeling serves as the canvas for boundless creativity. The ability to seamlessly blend mathematical precision with artistic vision underscores the transformative potential of geometric modeling in the realm of visual storytelling and entertainment. In addition to its technical and artistic facets, geometric modeling holds immense practical value in diverse fields such as architecture, industrial design, and medical imaging. Architectsleverage geometric modeling to conceptualize and visualize architectural designs, enabling them to explore spatial relationships, structural integrity, andaesthetic considerations. Similarly, industrial designers harness the power of geometric modeling to prototype and refine product designs, streamlining the development process and fostering innovation. Moreover, in the realm of medical imaging, geometric modeling plays a pivotal role in the reconstruction of anatomical structures from imaging data, enabling healthcare professionals to visualize and analyze complex biological forms with unprecedented clarity. The interdisciplinary nature of geometric modeling underscores its far-reaching impact, transcending the boundaries of technology, art, and practical application. As technological advancements continue to propel the evolution of geometric modeling, the boundaries of what can be visualized and created are continually expanding. The fusion of creativity, precision, and practicality inherent in geometric modeling exemplifies its enduring relevance in shaping the visual landscape of the future. Whether in the realm of virtual reality, architectural design, orscientific visualization, the profound influence of geometric modeling is poisedto continue shaping our perceptions and experiences in profound ways.。

促进人工智能大模型融合## Promoting the Integration of AI Large Models.Abstract.Artificial intelligence (AI) large models have shown remarkable progress in various domains, such as natural language processing, computer vision, and speech recognition. However, these models are often trained on specific datasets and tasks, which limits their generalization ability and applicability to real-world scenarios. To address this issue, promoting the integration of AI large models is crucial for unlocking their full potential. This article explores strategies and benefits of integrating AI large models, providing insights into the path forward for advancing this field.Strategies for Integrating AI Large Models.Integration of AI large models can be achieved throughseveral strategies:Data Fusion: Combining datasets from multiple sources can enhance the diversity and completeness of the data used to train large models, improving their generalization ability.Model Fusion: Merging multiple large models with different expertise can create a more comprehensive and robust model that leverages the strengths of eachindividual model.Knowledge Transfer: Transferring knowledge from pre-trained large models to smaller models or models for specific tasks can accelerate the training process and improve performance.Incremental Learning: Continuously updating and improving large models by incorporating new data and knowledge enables adaptation to changing environments and evolving tasks.Benefits of Integrating AI Large Models.Integrating AI large models offers numerous benefits:Enhanced Generalization: Fusion of diverse data and models enables large models to handle a broader range of tasks and scenarios, making them more versatile and applicable.Improved Performance: Combining complementary models and knowledge can boost accuracy, efficiency, and reliability, leading to better results across tasks.Reduced Training Time and Cost: Transferring knowledge and incrementally learning can significantly reduce the time and resources required to train large models.Accelerated Innovation: Integration fosters collaboration and knowledge sharing, stimulating the development of new and innovative AI applications.Conclusion.Promoting the integration of AI large models is vitalfor realizing their transformative potential. By leveraging diverse data, combining expertise, transferring knowledge, and enabling continuous learning, we can create more robust, versatile, and efficient models that drive advancements in various domains. As the field of AI continues to evolve,the integration of large models will play a pivotal role in shaping the future of AI-driven innovation.## 中文回答：促进人工智能大模型融合。

Geometric ModelingGeometric modeling is a fundamental concept in computer graphics and design, playing a crucial role in various industries such as architecture, engineering, and entertainment. It involves creating digital representations of physical objects or environments using mathematical and computational techniques. Geometric modeling allows designers and engineers to visualize, analyze, and manipulate complex shapes and structures, leading to the development of innovative products and solutions. However, it also presents several challenges and limitations that need to be addressed to ensure its effectiveness and efficiency. One of the key challenges in geometric modeling is the accurate representation of real-world objects and environments. This requires the use of advanced mathematical algorithms and computational methods to capture the intricate details and complexities of physical entities. For example, creating a realistic 3D model of a human face or a natural landscape involves precise measurements, surface calculations, and texture mapping to achieve a lifelike appearance. This level of accuracy is essential in industries such as animation, virtual reality, and simulation, where visual realism is critical for creating immersive experiences. Another challenge in geometric modeling is the efficient manipulation and editing of geometric shapes. Designers and engineers often need to modify existing models or create new ones to meet specific requirements or constraints. This process can be time-consuming and labor-intensive, especially when dealing with large-scale or highly detailed models. As a result, there is a constant demand for more intuitive and user-friendly modeling tools that streamline the design process and enhance productivity. Additionally, the interoperability of geometric models across different software platforms and systems is a persistent issue that hinders seamless collaboration and data exchange. Moreover, geometric modeling also faces challenges in terms of computational resources and performance. Generating and rendering complex 3D models requires significant computing power and memory, which can limit the scalability and accessibility of geometric modeling applications. High-resolution models with intricate geometries may strain hardware capabilities and lead to slow processing times, making it difficult for designers and engineers to work efficiently. This is particularly relevant in industries such as gamingand virtual reality, where real-time rendering and interactive simulations are essential for delivering engaging and immersive experiences. Despite these challenges, geometric modeling continues to evolve and advance through technological innovations and research efforts. The development of advanced modeling techniques such as parametric modeling, procedural modeling, and non-uniform rational B-spline (NURBS) modeling has significantly improved the accuracy and flexibility of geometric representations. These techniques enable designersand engineers to create complex shapes and surfaces with greater precision and control, paving the way for more sophisticated and realistic virtual environments. Furthermore, the integration of geometric modeling with other disciplines such as physics-based simulation, material science, and machine learning has expanded its capabilities and applications. This interdisciplinary approach allows for the creation of interactive and dynamic models that accurately simulate physical behaviors and interactions, leading to more realistic and immersive experiences. For example, in the field of architecture and construction, geometric modeling combined with structural analysis and environmental simulation enables the design and evaluation of sustainable and resilient buildings and infrastructure. In conclusion, while geometric modeling presents several challenges and limitations, it remains an indispensable tool for innovation and creativity in various industries. The ongoing advancements in geometric modeling techniques and technologies continue to push the boundaries of what is possible, enabling designers and engineers to create increasingly realistic and complex digital representations of the physical world. As computational power and software capabilities continue to improve, the future of geometric modeling holds great promise for revolutionizing the way we design, visualize, and interact with the world around us.。

大语言模型核心技术介绍Language models are a type of artificial intelligence system that has gained significant traction in recent years. These models are designed to process and generate human language, with the goal of improving natural language processing tasks such as translation, text generation, and question answering. Language models have become increasingly sophisticated, with the most recent advancements in large language models being some of the most significant in the field.语言模型是一种人工智能系统，近年来取得了显著的进展。

这些模型旨在处理和生成人类语言，以改进自然语言处理任务，如翻译、文本生成和问答。

语言模型变得越来越复杂，最近大语言模型的最新进展是该领域最重要的之一。

One of the key components of large language models is the use of neural networks. These networks are designed to mimic the way the human brain processes information, allowing them to learn from large amounts of data and improve their performance over time. By using neural networks, language models are able to understand thecomplex patterns and structures within human language, leading to more accurate and natural language generation.大型语言模型的关键组成部分之一是使用神经网络。

Geometric ModelingGeometric modeling is a crucial aspect of computer-aided design (CAD) and computer graphics. It involves the creation of digital representations of physical objects and shapes using mathematical equations and algorithms. Geometric modeling has a wide range of applications, including architecture, engineering, animation, and manufacturing. In this response, we will explore the importance of geometric modeling, its various techniques, and its impact on different industries. One of the key aspects of geometric modeling is its ability to accurately represent complex shapes and structures. This is particularly important in fields such as architecture and engineering, where precise measurements and design specifications are essential. Geometric modeling allows designers and engineers to create 3D models of buildings, machines, and other objects, enabling them to visualize and analyze their designs before they are built. This not only helps to identify potential issues and flaws early in the design process but also saves time and resources by reducing the need for physical prototypes and revisions. In addition to its practical applications, geometric modeling also plays a significant role in the entertainment and gaming industries. 3D modeling software is used to create realistic characters, environments, and special effects in movies, television shows, and video games. The ability to accurately represent and manipulate complex shapes and textures is essential for creating immersive and visually stunning digital experiences. Geometric modeling techniques such as subdivision surfaces and NURBS (Non-Uniform Rational B-Splines) are commonly used to create smooth and detailed 3D models in the entertainment industry. Furthermore, geometric modeling has revolutionized the field of manufacturing. With the advancement of technologies such as 3D printing and computer numerical control (CNC) machining, geometric models are used to create physical objects with a high degree of precision and complexity. Manufacturers can use 3D models to prototype and produce custom parts and products, leading to greater flexibility and efficiency in the production process. Geometric modeling also enables the optimization of designsfor additive manufacturing, allowing for the creation of lightweight and structurally efficient components. From a mathematical perspective, geometric modeling involves the use of various mathematical concepts and techniques torepresent and manipulate shapes and objects. This includes the use of coordinate systems, transformations, and equations to describe the position, orientation, and geometry of 3D objects. Geometric modeling also encompasses the use of algorithms for tasks such as rendering, mesh generation, and surface fitting. These mathematical foundations are essential for the development of efficient and accurate geometric modeling software and algorithms. In conclusion, geometric modeling is a fundamental aspect of modern design, engineering, and entertainment. Its ability to accurately represent and manipulate 3D shapes and objects has revolutionized various industries, leading to advancements in design, manufacturing, and digital media. The mathematical principles and techniques behind geometric modeling are essential for the development of sophisticated software and algorithms. As technology continues to advance, geometric modeling will continue to play a vital role in shaping the way we design and create in the digital age.。

Geometric ModelingGeometric modeling is a crucial aspect of computer graphics and design,playing a fundamental role in creating virtual representations of physical objects and environments. It involves the mathematical representation of shapes, surfaces, and volumes, allowing for the visualization, analysis, and manipulation of complex structures. From architectural design and industrial engineering to animation and video game development, geometric modeling is widely used across variousindustries and applications. One of the primary challenges in geometric modelingis achieving a balance between accuracy and efficiency. On one hand, the models need to be precise and faithful to the real-world objects they represent. On the other hand, the computational cost of processing and rendering these models should be manageable, especially in real-time applications such as interactivesimulations and virtual reality environments. This trade-off often requirescareful consideration of the level of detail, the choice of representation (e.g., polygonal, parametric, or procedural), and the optimization of algorithms for geometric operations. Another key issue in geometric modeling is the handling of geometric complexity. Many real-world objects have intricate shapes and intricate structures that can be challenging to capture and manipulate in a digital environment. This complexity can arise from organic forms in nature, irregular patterns in architecture, or detailed surface textures in industrial design. Addressing this challenge involves the development of advanced modeling techniques, such as subdivision surfaces, non-uniform rational B-splines (NURBS), and level-of-detail (LOD) representations, as well as the use of specialized tools for sculpting, texturing, and displacement mapping. Furthermore, geometric modeling often intersects with other disciplines, such as computer-aided design (CAD), computational geometry, and physical simulation. This interdisciplinary nature introduces additional considerations, such as the compatibility of geometric data across different software platforms, the integration of geometric constraints and parametric modeling, and the incorporation of physical properties and behaviorsinto the models. These interactions highlight the importance of interoperability, standardization, and collaboration among professionals working in related fields. From a practical standpoint, the evolution of geometric modeling has been drivenby advancements in hardware and software technologies. The increasingcomputational power of modern computers has enabled the handling of larger andmore detailed geometric datasets, while the development of specialized graphics processing units (GPUs) has accelerated the rendering and visualization of complex3D models. Similarly, the availability of sophisticated modeling software, such as Autodesk Maya, Blender, and Rhinoceros, has empowered designers and artists to explore new creative possibilities and push the boundaries of geometric expression. In conclusion, geometric modeling encompasses a broad range of challenges and opportunities in the realm of computer graphics and design. It requires a delicate balance between accuracy and efficiency, the ability to handle geometric complexity, and a deep understanding of interdisciplinary connections. Astechnology continues to advance, the future of geometric modeling holds promisefor further innovation and creativity, shaping the way we perceive and interactwith virtual representations of the world around us.。

JAN SWEVERS, WALTER VERDONCK, and JORIS DE SCHUTTERINTEGRATED EXPERIMENT DESIGN AND PARAMETER ESTIMATIONIndustrial robot manipulators are indispensable for achieving productivity and flex ibility in fully auto-mated production lines, where they are used for a wide variety of tasks, ranging from material handling and assembly to cutting, welding, gluing, and painting. To improve productivity and accuracy, robot manufac-turers invest time and effort in developing advanced offline programming tools and controllers. Dynamic robot models are a key element in these developments.OFFLINE PROGRAMMING AND TASK OPTIMIZATIONIn industrial practice, many robots are programmed manually by leading the robot through a sequence of position points. This approach, known as online programming, has the advan-tage that positions defined by task execution are more accurate than a path specified in a software environment. On the other hand, online programming requires idle timethat may be unacceptable in the case of small product batch sizes.Robot setup times can be kept short if the task is programmed without interrupting the production process. Thisapproach, known as offline programming,occurs in a softwareDynamic Model Identification for Industrial Robots58IEEE CONTROL SYSTEMS MAGAZINE»OCTOBER 20071066-033X/07/$25.00©2007IEEENASA/GODDARD SPACE FLIGHT CENTER SCIENTIFIC VISUALIZATION STUDIODigital Object Identifier 10.1109/MCS.2007.904659environment. Many rob ot manufacturers, such as KUKA, ABB, and Fanuc, as well as third-party companies, offer software for offline programming. These interactive simu-lation environments allow the user to create virtual models of production environments and to program and simulate robot manipulator tasks before transferring the code to the robot controller (Figure 1). In particular, three-dimensional graphics are used to coordinate the elements of the pro-duction process and detect possib le collisions and other system failures.Offline programming software does not consider uncertainty in the position and orientation of the ob jects in the production environment. In addition, it is assumed that the robot manipulator executes the programmed task with perfect position accuracy. However, the trajectories executed b y the rob ot manipulator in practice differ sig-nificantly from the programmed and simulated trajecto-ries due to kinematic modeling errors and rob ot dynamics. Inaccuracies due to kinematic errors can b e compensated for by using a calibrated kinematic model of the rob ot manipulator [1]. However, for high-speed motions, or in case of a heavy payload, the tracking error is caused mainly by dynamic forces, for example, centrifu-gal and Coriolis forces, dynamic coupling b etween the joint axes, and actuator dynamics. These effects are not sufficiently accounted for b y standard industrial rob ot controllers. Dynamic effects can be compensated, howev-er, b y switching to more advanced model-b ased con-trollers or b y adapting the programmed rob ot trajectory. In the latter case, the offline programming environment must include dynamic rob ot models as well as dynamic-simulation functionality.Another objective of offline programming is to minimize the cycle time of a robot task, since shorter cycle times lead to higher productivity. However, physical constraints, such as maximum motor and gearbox torques as well as maxi-mum motor speeds, must be accounted for to avoid over-load, which causes accelerated wear and tear of the actuators, gears, and bearings, resulting in loss of accuracy and even premature failure. Realistic simulation and opti-mization of the motion of a robot manipulator, including its physical constraints, requires an accurate dynamic model. Advanced Robot ControlStandard industrial robot controllers consist of independent proportional-integral-differential (PID)-like position con-trollers, one for each joint. Although these controllers yield sufficiently accurate path tracking for most industrial appli-cations, applications such as laser welding and laser cutting require higher accuracy. These applications are character-ized by complex six-degree-of-freedom trajectories, fast motion, and stringent path-tracking accuracy requirements. Fast motion results in high dynamic coupling between the various robot links, which cannot be compensated for by a standard robot controller. Consequently, advanced robot controllers have been developed, for example, computed-torque control as well as feedforward dynamic compensa-tion [2]–[5]. These controllers are more complex than standard industrial robot controllers since they are based on a model of the complete robot dynamics. Although practical industrial conditions are far from those in laboratory envi-ronments, results obtained on industrial setups confirm the benefits of model-based control [6]. Improved tracking accu-racy is obtained, provided that the robot model is sufficient-ly accurate for torque prediction.INTEGRATED EXPERIMENTALROBOT IDENTIFICATIONA dynamic rob ot model, which relates rob ot motion to joint torques, describ es the rigid-b ody dynamics of the rob ot and includes Coulomb and viscous friction in the joints. Although inertia estimates can b e derived from CAD drawings, robot manufacturers do not provide these drawings for all parts of the robot, for example, parts man-ufactured b y external suppliers. Dismantling the rob ot to measure mass and inertia properties of the links is not a realistic option. Moreover, estimates of friction parameters are not provided b y the manufacturers and are not pre-dictable from first principles.Experimental identification using motion and torque data measured during experiments is thus needed to ob tain accurate estimates of rob ot model parameters. In particular, we present an experimental robot identification procedure b ased on the maximum likelihood framework with periodic bandlimited excitation [7].The input to the identification procedure is kinematic and geometric information about the robot manipulator, as well as specifications about the desired model accuracy; seeFIGURE 1 Illustrative graphical output of the KUKA offline program-m ing environm ent KUKASim. A com plete production process can be sim ulated in a virtual environm ent, collisions can be detected, and robot program s can be generated autom atically. Cycle-tim e optim ization is possible but is lim ited to geom etric and kinem atic information. (Courtesy of KUKA.)OCTOBER 2007«IEEE CONTROL SYSTEMS MAGAZINE59Figure 2. These inputs determine the choices to be made in the procedure. For example, kinematic and geometric infor-mation about the robot includes the number of joints, the orientations of the joint axes, and the lengths of the robot links. Model accuracy specifications determine the type of model to be used and the level of detail of the dynamics to be included in the model. These choices have implications for the various steps of the identification procedure.The last step of the identification procedure is model validation, where the user verifies that the model satisfies the accuracy specifications. If the obtained model does not pass the validation tests, one or several steps of the proce-dure are repeated and some of the choices are reconsidered.MODELINGDynamic robot models define the relationship between the motion of the robot manipulator and the actuator torques.The motion of the robot is described by the position, veloc-ity, and acceleration of all of its links. The robot is repre-sented by a kinematic chain of rigid bodies, and thus rigid-body dynamic e quations are the basis for the mode l.Depending on the system and on the specifications, these rigid-body equations have to be complemented with mod-els of other effects such as friction and gravity-compensat-ing devices if present.Rigid-Body DynamicsThe Newton-Euler or Lagrangian method is used to derive the dynamic equations of kinematic chains of rigid bodies [8]. Both approaches yield the dynamics equationτ=M (q ,ϑ)¨q+C (q ,˙q ,ϑ)+g (q ,ϑ),(1)which e xpre sse s, for an n -de gre e -of-fre e dom robot, the n -vector of actuator torques τas a function of the n -vectorsof the joint positions q , velocities ˙q, and accelerations ¨q as well as the model parameters ϑ. In (1), M (q ,ϑ)is the n ×ninertia matrix, C (q ,˙q,ϑ)is the n -vector containing Coriolis and centrifugal forces, and g (q ,ϑ)represents gravitationaltorques. M (q ,ϑ),C (q ,˙q,ϑ), and g (q ,ϑ)are nonlinear func-tions of the model parameters ϑ, that is, the mass, center-of-gravity location, and mome nts and products of ine rtia of each link.Using the baryce ntric parame te rs [9] or the modifie d Newton-Euler parameters [8] yields a model of the formτ= (q ,˙q,¨q )θ,(2)which is linear in the unknown parameters. In (2), θis the barycentric parameter vector, and is the observation or ide ntification matrix, which de pe nds only on the motion data. This prope rty simplifie s the parame te r e stimation considerably [10]. The barycentric parameters of a link are combinations of the inertial parameters of the link and its descendants in the kinematic chain [11]. For example, the baryce ntric mass of a link is de fine d as the mass of that link augmented by the total mass of all descendant links.The modifie d Ne wton-Eule r parame te rs e xpre ss the ine r-tial parame te rs as first- and se cond-orde r mome nts with respect to a link frame located at the joint axis.Gravity Compensation, Dynamic Coupling, and FrictionRigid-body equations (1) and (2) include only the effects of link masses and inertias. Friction, dynamic coupling due to the inertia of geared actuator rotors that spin at high veloc-ity [12], as we ll as the e ffe cts of gravity-compe nsating de vice s if pre se nt contribute significantly to the dynamic be havior of the robot manipulator. Gravity-compe nsating de vice s, which are pre loade d springs mounte d be twe e n the first and se cond link, approximate ly compe nsate the static torque caused by the mass of the payload and robot wrist on the actuator of the second link, that is, the shoul-der actuator [13]. Dynamic coupling and gravity-compen-sating springs can be de scribe d by mathe matical60IEEE CONTROL SYSTEMS MAGAZINE»OCTOBER 2007FIGURE 2 Schemati c representati on of a standard experi mentalrobot i denti fi cati on procedure. The ki nemati c and geometri c i nfor-mation of the robot manipulator and model-accuracy specifications are the i nputs to the i denti fi cati on procedure. Thi s i nformati on i s avai lable pri or to the i denti fi cati on and determi nes choi ces to be made i n the procedure. The model vali dati on step evaluates the accuracy of the model according to criteria that depend on the appli-cation of the model. If the identified model does not pass the valida-tion tests, one or several steps of the procedure are repeated and choices are reconsidered.expressions that are linear in the parameters. Consequent-ly, these effects fit conveniently within the linea r-in-pa ra-meters model structure (2).Although friction is a complex nonlinear phenomenon, especia lly during motion reversa l [14], [15], a friction model consisting of only Coulomb a nd viscous friction, that is,τf ric=f C sign(˙q)+f v˙q,(3)is a n a ccepta ble simplifica tion for ma ny robotics a pplica-tions [16]. Note that (3) is linear in the unknown Coulomb and viscous friction parameters f C and f v and thus is con-sistent with the linear model structure (2).EXPERIMENT DESIGNDuring the design of an identification experiment, it is nec-essary to ensure that the excitation is sufficient to provide a ccura te a nd fa st pa ra meter estima tion in the presence of disturbances, and that the processing of the resulting data is simple a nd yields a ccura te results. The experiment design consists of two steps. First, a trajectory parameteri-zation is selected, and second the trajectory parameters are calculated, usually by means of optimization.Trajectory ParameterizationSeveral approaches exist for parameterizing robot-excitation trajectories, for example, finite sequences of joint accelera-tions [17], or fifth-order polynomials interpolating between sets of joint positions and velocities separated in time [18]. Although these trajectories provide adequate excitation of the robot dynamics, the resulting measurement data are nei-ther periodic nor bandlimited. Processing periodic, ban-dlimited measurements is more accurate, simplifying and improving the accuracy of the parameter estimates.Periodic bandlimited measurements are obtained if the excitation is periodic and bandlimited, that is, if the trajec-tory q i(t)of ea ch joint i is periodic, pa ra meterized a s a finite Fourier seriesq i(t)=q i,0+Nk=1(a i,k sin(kωf t)+b i,k cos(kωf t)),(4)where t represents time, and ωf is the same for all joints. This Fourier series is periodic with period T f=2π/ωf, which is chosen to be an integer multiple of the sampling period. Ea ch Fourier series conta ins 2N+1parameters, which are the degrees of freedom for trajectory optimiza-tion. The coefficients a i,k and b i,k are the amplitudes of the sine a nd cosine functions a long with the offset q i,0of the position trajectory.In choosing the frequency range [ωf,Nωf]of the exci-tation trajectories, the following trade-off has to be consid-ered. By selecting a low fundamental frequency ωf, that is, a long excita tion period, a la rger pa rt of the robot work-spa ce ca n be covered for given ma ximum joint velocities, however, a t the cost of longer mea surement time. Good coverage of the robot workspace improves the information content of the measurements as well as the accuracy of the pa ra meter estima tes [19]. On the other ha nd, including high frequencies provides high a ccelera tions, which a re required to accurately estimate the moments and products of inertia. The highest frequency of the commanded trajec-tory, however, is limited by the lowest resonance frequen-cy of the robot structure. The structural flexibilities of the robot are excited if the highest frequencies of the trajectory are too close to the lowest resonance frequency. Excitation of flexible modes is disadvantageous because they are not accounted for in the rigid-body robot model. As discussed below, the comma nded tra jectory need not be followed exactly since the actual motion of the robot is measured. Trajectory OptimizationAppropria te va lues for the tra jectory pa ra meters ca n be selected either by trial and error or by solving a nonlinear optimiza tion problem with constra ints imposed on the robot motion.Several objective functions exist for trajectory optimiza-tion. A popula r optimiza tion criterion is the loga rithm of the determina nt of the cova ria nce ma trix of the model pa ra meter estima tes, known a s the d-optimality criterion [10]. This criterion mea sures the size of the uncerta inty region of the model pa ra meter estima tes. Its ca lcula tion does not depend on the model parameters if the joint posi-tion, velocity, a nd a ccelera tion da ta a re free of noise but ra ther depends only on the robot tra jectory through the identification matrix in (2) as well as on the covariance of the noise on the a ctua tor torque mea surements [20]. This property is useful in practice since the optimization of the robot excita tion ca n be performed without a ny prior knowledge of the model parameters.The motion constra ints impose limita tions on the joint positions, velocities, a nd a ccelera tions a s well a s on the robot end-effector position in Cartesian space. These limi-tations avoid collisions between the robot and objects in its environment as well as collisions between robot links.OCTOBER 2007«IEEE CONTROL SYSTEMS MAGAZINE61ROBOT EXCITATION, DATA ACQUISITION, AND SIGNAL PROCESSINGThe optimized robot-excitation trajectory is programmed in the robot controller. The robot repeats the trajectory contin-uously, while data are collected at a constant, user-specified sampling frequency. Data collection starts after the transient response caused by the startup of the experiment dies out.Joint positions are measured using encoders mounted on the actuator shafts. Since the measured joint trajecto-ries deviate from the desired trajectories, due to the limi-tations of the robot control l er, the measured joint positions are used for parameter estimation instead of thecommanded trajectories. Al though these measurements may contain more or different harmonics than the com-manded trajectories, the motion is periodic, with the same period as that of the commanded trajectories, and ban-dlimited, because the robot is controlled by a bandlimited position-feedback controller.The actuator torque data are obtained through actuator current measurements without additional sensors. The relationship between current and torque is modeled as lin-ear or as a higher order polynomial [21], of which the para-meters are provided by the motor manufacturer or identified in a separate experiment [22].The aim of the signal processing step is to clean up the measured data. This step improves the signal -to-noise ratio of torque and joint position measurements, estimates the variance of the measurement noise, and calculates the joint velocity and acceleration estimates based on the mea-sured joint positions.Data Averaging and Noise Variance EstimationMeasurement noise, which is assumed to be an additivenormal l y distributed zero-mean stochastic disturbance,causes uncertainty and bias errors in the parameter esti-mates. For a given data set, bias errors can be avoided and uncertainty can be minimized by using an efficient estimator, for exampl e, a maximum l ikel ihood estima-tor. This estimator yiel ds unbiased parameter estimates with minimal uncertainty when correct val ues for the noise covariance are used [19, pp. 20–25] and [23, pp.92–106].Since the data are periodic, the signal-to-noise ratio can be improved by data averaging without using a l owpass noise filter. Averaging, which is used by all spectrum ana-l yzers, improves the qual ity of the data with the square root of the number of measured periods.To estimate the noise l evel on measured periodic sig-nals, the sample variance of a signal x consisting of M peri-ods of K samples is given byσ2x=1(MK −1)K k =1M m =1(x m (k )−¯x(k ))2,(5)where x m (k )indicates the k th sample within the m th peri-od, and ¯x(k )denotes the average of x , that is, ¯x (k )=1M Mm =1x m (k ).(6)FIGURE 3 Exact frequency-domain differentiation of a measured peri-odic joint position signal. The left column explains the various steps of the procedure. The right column illustrates these steps for a peri-odic signal containing only one frequency, that is, a sinusoidal sig-nal, perturbed with ad d itive normally d istributed noise. First, the signal is transformed to the frequency d omain using the d iscrete Fourier transform (DFT). Next, the spectrum is multiplied by a rec-tangular window, which selects the frequencies that contain signal information. For this example, the multiplication correspond s to selecting one frequency. The spectrum is set to zero at all other fre-quencies. The resulting spectrum is then multiplied by the continu-ous-time frequency-domain representation of a differentiator at the selected frequency. That is, the spectrum is multiplied by j ω(k )=j 2πkf s /P , with f s the sampling frequency, P the number of time-domain samples, k ={−1,1}is the index of the selected fre-quency in the discrete spectrum. A transformation back into the time d omain using the inverse DFT yield s an estimate of the first time derivative of the original signal, that is, the velocity. The velocity sig-nal is almost free of noise, that is, noise is removed from all frequen-cies except the selected ones. For simplicity, only the amplitud e spectra are shown.62IEEE CONTROL SYSTEMS MAGAZINE»OCTOBER 2007Fo r ro bo t identificatio n the signal x co rrespo nds to the measured actuator torques τand joint positions q.Exact Calculation of the Joint Velocitiesand AccelerationsCalculatio n o f the identificatio n matrix given by (2) requires estimates o f the jo int velo cities and accelera-tio ns. Numerical differentiatio n amplifies the no ise pre-sent in the measurements, while noisy joint velocity and acceleratio n estimates deterio rate the accuracy o f the parameter estimates.Exact differentiation is possible when the joint position measurements are periodic, bandlimited, and the sampling frequency f s is at least twice the bandwidth to avoid alias-ing. To calculate the joint velocities and accelerations, the averaged jo int po sitio n measurements are transfo rmed to the frequency domain using the discrete Fourier transform. This transformation does not introduce leakage errors if an integer number of periods is selected. In this case, the cal-culated spectrum is exactly equal to the co ntinuo us-time Fourier transform, which consists of discrete components in the frequency do main and co rrespo nds to the Fo urier series of that signal.Next, the relevant frequencies are selected by frequency-do main windo wing, using a rectangular windo w, that is, the spectrum is set to zero at all but the selected frequen-cies, which co rrespo nds to frequency-do main data filter-ing. The selected spectrum is then multiplied by the co ntinuo us-time frequency respo nse o f a pure single and do uble differentiato r to o btain velo city and acceleratio n estimates. That is, the spectrum is multiplied by jω(k)and −ω(k)2, respectively, where ω(k)=2πkf s/P, P is the num-ber of samples of the signal, and k is the set of indices of selected frequencies in the discrete spectrum (discrete Fo urier transfo rm). The resulting spectrum is then trans-formed back to the time domain using the inverse discrete Fo urier transfo rm. Figure 3illustrates this o peratio n fo r velo city estimatio n. This frequency-do main data filtering, which is also applied to the po sitio n measurements, remo ves the no ise fro m all but the selected frequencies, yielding accurate, fo r practical purpo ses no ise-free, po si-tion, velocity, and acceleration signals. The results are not perfectly free o f no ise because no ise canno t be remo ved from the selected frequencies.PARAMETER ESTIMATION AND MODEL VALIDATION The selectio n o f a parameter-estimatio n metho d is a compromise between accuracy and complexity of imple-mentatio n. Linear least squares parameter estimatio n (LLSE), which sits at one side of this spectrum, is a non-iterative method that finds the parameter estimates in a single step using the singular value deco mpo sitio n. LLSE do es no t discriminate between accurate and inac-curate data, and thus yields biased estimates with no n-minimal uncertainty.The maximum likeliho o d estimatio n sits at the o ther side o f the spectrum since it pro vides unbiased estimates with minimal uncertainty regardless of the spectrum of the measurement no ise. The maximum likeliho o d estimate o f the parameter vector θis the value of θthat maximizes the likelihood of the measurements. This criterion, which is a no nco nvex functio n o f the unkno wn mo del parameters, depends o n the co variance o f the no ise o n all measure-ments and derived variables such as jo int velo cities and acceleratio ns. So lving this no nlinear least squares o pti-mization problem is often cumbersome because it requires an initial guess o f the parameters and because it might converge to a local optimum, yielding a suboptimal solu-tion that may be biased.When the joint position, velocity, and acceleration data are free o f no ise, which is fo r practical purpo ses realized by applying the abo ve-mentio ned signal pro cessing, the identification matrix in (2) is free of noise, and the maxi-mum likeliho o d estimatio n simplifies to weighted linear least squares estimation (WLSE). The complexity of WLSE is comparable to that of LLSE, in fact, the only difference is that WLSE weighs the data with the inverse of the covari-ance of the actuator torque measurement noise and there-OCTOBER 2007«IEEE CONTROL SYSTEMS MAGAZINE63 FIGURE 4A KUKA IR 361 industrial robot in the robotics laboratory at KU Leuven. Th e position controller of th e robot is provided by Orocos software [24], [25]. Th is software facilitates th e application of periodic commanded trajectories, wh ile synch ronizing encoder and actuator current measurements.f ore discriminates between accurate and inaccurate data.WLSE has the same favorable properties as the maximum likelihood estimation.The weighted least squares estimate of the model para-meters θisˆθWLS =(F T −1F )−1F T −1τ(7)=( −1/2F )+ −1/2τ,(8)where (·)+denotes the pseudoinverse,F =⎡⎢⎣ (q t 1,˙qt 1,¨q t 1)...(q t K ,˙qt K ,¨q t K )⎤⎥⎦,(9)τ=⎡⎢⎣τt 1...τt K⎤⎥⎦.(10)Here, (q t k ,˙qt k ,¨q t k )and τt k , for k =1,...K , are the identifi-cation matrix and torque vector evaluated using the joint position, velocity, and acceleration data, as well as the actuator torque data at the discrete-time instant t k . Further-more, is the covariance matrix of the actuator torque data, which is a dense matrix if the noise is correlated.Here, the assumption is made that the noise is white,which is made for convenience. Since the matrix F in (9) is free of noise, the parameter estimates are unbiased even if the actual noise is colored, although the estimates may have nonminimal uncertainty [23, pp. 92–106].The covariance matrix of the estimated parameter vec-tor ˆθWLS is C =(F T −1F )−1,(11)which, along with the related uncertainty bounds on the parameters, can be used for model validation.The aim of the model-validation step is to obtain confi-dence in the estimated robot model in view of its intended application. Obviously, the most appropriate validation test is to use the model in the application and evaluate its success. This validation method, however, can lead to undesirable and even dangerous situations. Theref ore,validation must be addressed prior to the actual applica-tion. Given this requirement, a validation test must evalu-ate the properties of the estimated model and the parameter estimates that are relevant to the intended application. Unsatisfactory validation leads to a reconsid-eration of some previous steps of the identification proce-dure, f or instance, a new experiment design or a more elaborate dynamic model.We consider two model-validation measures, the actua-tor torque-prediction accuracy and the parameter-estima-tion accuracy.Actuator Torque-Prediction AccuracyThe actuator torque-prediction accuracy of a robot model is needed f or of f line programming, task opti-mization, and advanced robot control. The robot model (2) is evaluated for some desired motion, described by aset of joint positions q , velocities ˙q, and accelerations ¨q ,FIGURE 5 Optimized robot-exc itation trajec tory. One period of the optimized joint trajectories is shown. These trajectories consist of a five-term Fourier series with a base frequency of 0.1 Hz. The trajec-tory parameters are optimized according to the d -optimality criterion,taking into account workspace limitations, constraints on joint veloci-ties, and ac c elerations. The optimization c riterion is a measure of the size of the overall uncertainty region of the parameter estimates.64IEEE CONTROL SYSTEMS MAGAZINE»OCTOBER 2007。

重庆大学硕士学位论文英文摘要ABSTRACTTwo-wheeled differentially driven mobile robot (TDDMR) has simple structure and is easy control. The study of robot motion model is an important part of academic field and is of practical significance for accurate description and control of mobile robot.In physical robot system, there have the following characteristics:①The drive system always contains two double closed-loop control structurewhere nonlinear links exist according to previous study. Meanwhile, Changes in loads of the robot also influence the dynamic behavior of drive system.② Two-wheeled differentially driven mobile robot is a system subject to classical nonholonomic constraints.③ As a typical multi-input-multi-output (MIMO) system, the mobile robot system has mutual coupling between two underlying control circuit.In a word, the mobile robot is not only subject to nonholonomic constraints, but also a multi-input-multi-output (MIMO) system that contain closed-loop nonlinear links and dynamic coupling. Thus the establishment of the robot motion model is an emphasis and difficulty in theoretic study related to nonlinearity.The quasi-equivalent modeling approach is an nonlinear modeling method by which some important parameters can be included in the model that reflect major properties of physical system . This approach is adopted to establish the motion model of the TDDMR . The main works are as follows:①Based on double closed-loop control system load variable quasi-equivalent model, the state space motion model of two-wheeled differentially driven mobile robot is established.By introducing the dynamic coupling into dynamic structure of double-closed loop drive system and converting the loads of robot into equivalent moment of inertia of the model, the quasi-equivalent state space motion model is obtained by means of the quasi-equivalent modeling approach.②The parameter tuning method of quasi-equivalent state-space motion model based on genetic algorithm is proposed.The robot system consists of two drive system coupled with each other, so the improved genetic algorithm is used to tune parameters of the model by applyingII重庆大学硕士学位论文英文摘要simultaneously step signals to the two drive system.③To validate the motion model and parameter tuning method, a novel load-adjustable two-wheel differential drive mobile robot being the experiment platform is designed in this paper .Experimental findings show that the modeling method and motion model proposed in this paper prove as practical.Keywords：Differentially Driven Mobile Robot, Quasi-equivalent Modeling, Genetic Algorithms, Equivalent Moment of InertiaIII重庆大学硕士学位论文目录目录中文摘要........................................................................................................ I 英文摘要....................................................................................................... II 1 绪论 (1)1.1 课题提出的背景 (1)1.2 国内外研究现状 (2)1.3 存在的问题和研究意义 (4)1.3.1 存在的问题 (4)1.3.2 研究意义 (8)1.4 本文的研究内容及结构安排 (8)2 负载可调轮式移动机器人 (9)2.1 引言 (9)2.2 移动机器人机械结构 (9)2.3 机器人控制系统硬件 (11)2.3.1 DSP控制器单元 (12)2.3.2 电机及驱动器单元 (13)2.3.3 扩展电路 (16)2.4 机器人控制系统软件 (18)2.4.1 通信协议设计 (19)2.4.2 任务设计 (20)2.5 本章小结 (20)3 移动机器人类等效状态空间运动模型 (21)3.1 引言 (21)3.2 移动机器人驱动系统动力学模型 (21)3.2.1 移动机器人运动学方程 (21)3.2.2 移动机器人动力学方程 (22)3.2.3 移动机器人驱动系统动力学模型 (26)3.3 类等效模型简化 (27)3.3.1 类等效建模方法 (27)3.3.2 类等效简化过程 (29)3.3.3 模型状态空间表达式 (33)3.3.4 类等效状态空间运动模型结构 (35)IV重庆大学硕士学位论文目录3.4 本章小结 (36)4 两轮差速驱动移动机器人运动模型参数辨识及实验 (37)4.1 引言 (37)4.2 基于遗传算法的模型参数辨识 (37)4.2.1 标准遗传算法 (38)4.2.2 改进遗传算法 (38)4.3 适应度函数选取 (39)4.4 左右电机模型参数的辨识 (40)4.5 本章小结 (42)5 实验对比 (43)5.1 引言 (43)5.1.1 固定轮速下对比实验 (44)5.1.2 固定负载下对比实验 (48)5.1.3 点镇定控制下的对比实验 (52)5.2 本章小结 (55)6 结论与展望 (56)致谢 (57)参考文献 (58)附录 (61)A. 作者在攻读硕士学位期间发表论文目录 (61)V重庆大学硕士学位论文 1 绪论1绪论1.1课题提出的背景根据联合国教科文组织的定义，机器人是指一种可依据不同任务进行编程，能完成某种操作的专门系统。