Overview This repository is dedicated to the exploration and implementation of advanced control strategies for two critical areas of robotics and control engineering: Flexible Link Manipulator Modeling (also in a rotating reference frame), and Control. With a focus on adaptive, robust, and intelligent control methods, our goal is to push the boundaries of what's possible in the precision and adaptability of these complex systems. Utilizing MATLAB, this project covers comprehensive modeling, simulation, and control strategies designed to tackle the unique challenges presented by the flexibility and dynamics of these systems. Flexible Link Manipulators Flexible Link Manipulators (FLMs) represent a significant area of research in robotics, where the flexibility of the manipulator links is considered in the design of control systems. These systems pose unique challenges due to their inherent non-linear dynamics and the presence of significant vibrations and deflections during operation. In addition, the control of rotating beams involves managing the dynamics of beams under rotational motion, a common scenario in aerospace and mechanical engineering applications. This repository includes: • Modeling Techniques: Detailed MATLAB scripts and Simulink models capturing the complex dynamics of FLMs, including Lagrangian and Finite Element Methods for accurate physical representations. • Control Strategies: Implementation of adaptive, robust, and intelligent control algorithms to address the precision and stability challenges in FLM operations. Techniques include but are not limited to PID, Sliding Mode Control, and Neural Network-based controllers. • Dynamic Modeling: Development of mathematical models that accurately represent the behavior of rotating beams, considering factors like centrifugal stiffening and gyroscopic effects. • Advanced Control Methods: Application of state-of-the-art control techniques to ensure stability and performance under varying operational conditions. This includes adaptive control to automatically adjust to changing dynamics and robust control to maintain performance under uncertainty. Control Methods Our work emphasizes the development and implementation of various control methodologies tailored to the specific needs of FLMs and rotating beams: • Adaptive Control: Dynamically adjusting controller parameters in real-time for optimal performance. • Robust Control: Ensuring system stability and performance despite model uncertainties and external disturbances. • Intelligent Control: Leveraging artificial intelligence (AI) and machine learning (ML) techniques for predictive modeling and control, enhancing the system's ability to deal with complex, nonlinear dynamics efficiently. Getting Started This repository provides MATLAB code, Simulink models, and detailed documentation for each project. Whether you're a student, researcher, or industry professional, you'll find valuable resources here to advance your work in control engineering. • Prerequisites: MATLAB (with Control System Toolbox and Simulink), basic understanding of control theory and dynamics. • Installation: Clone this repository and follow the setup instructions in each project folder to get started. Contributing We welcome contributions from the community! Whether it's adding new control strategies, enhancing the models, or improving the documentation, your input is valuable. License This project is licensed under the CC BY 4.0.
flexiblelink_analytical.m
This MATLAB script is an analytical tool designed to investigate the mode shapes of a clamped-free beam subjected to an applied moment at the free end, leveraging the principles of Euler-Bernoulli beam theory. The script makes use of MATLAB's symbolic computation capabilities, allowing for the derivation and solution of the beam's differential equation under specific boundary conditions. Here's a detailed explanation tailored for both students and experts in the field:
Script Breakdown Symbolic Variables and Parameters matlab Copy code syms w(x) C1 C2 C3 C4 EI L M0 assume(EI > 0); assume(L > 0); assume(M0, 'real'); The script starts by defining symbolic variables for the beam's deflection w(x), integration constants C1 to C4, the beam's flexural rigidity EI, length L, and an external applied moment M0 at the beam's free end. Assumptions are set to ensure physical feasibility: EI and L are positive, and M0 is a real number, reflecting realistic conditions of a beam.
Differential Equation Formulation matlab Copy code diffEq = EI*diff(w, x, 4) == 0; This line formulates the differential equation based on Euler-Bernoulli beam theory, which states that the product of the beam's flexural rigidity EI and the fourth derivative of the deflection w(x) with respect to the beam's longitudinal axis x equals zero, under the assumption of no distributed load.
Solving the Differential Equation matlab Copy code wSol(x) = dsolve(diffEq); The differential equation is solved using MATLAB's dsolve function. This step provides a general solution wSol(x) that includes arbitrary constants C1 through C4, which will later be determined by applying specific boundary conditions.
Applying Boundary Conditions matlab Copy code BCs = [subs(wSol, x, 0) == 0, ... subs(diff(wSol, x), x, 0) == 0, ... subs(EIdiff(wSol, x, 2), x, L) == M0, ... subs(EIdiff(wSol, x, 3), x, L) == 0]; Boundary conditions for a clamped-free beam are applied here. At the clamped end (x=0), the deflection and its first derivative (slope) are zero, reflecting no displacement or rotation. At the free end (x=L), the moment equals M0, and shear force (third derivative of w) is zero. These conditions are crucial for finding the actual physical behavior of the beam under given constraints.
Solving for Constants matlab Copy code constants = solve(BCs, [C1, C2, C3, C4]); This step solves for the constants C1 through C4 using the previously defined boundary conditions, effectively customizing the general solution to the specific problem at hand.
Substituting Constants into the General Solution matlab Copy code modeShape = subs(wSol, constants); Substitutes the solved values of the constants back into the general solution, yielding the specific mode shape of the beam under the defined boundary conditions and applied moment.
Displaying and Plotting the Mode Shape matlab Copy code disp('Mode Shape Equation with Modified Boundary Conditions:'); disp(modeShape);
fplot(subs(modeShape, {EI, L, M0}, {69e9 * (6.35e-3)^4 / 12, 1, 10}), [0, 1]); Finally, the script displays the derived mode shape equation and plots it for given values of EI, L, and M0. This visual representation aids in understanding how the beam deflects along its length due to the applied moment at the free end.
Tailored Explanation For Students: This script is a practical demonstration of applying theoretical knowledge from structural mechanics to solve real-world engineering problems using computational tools. It shows how to transition from the abstract formulation of Euler-Bernoulli beam theory to a concrete solution that describes the physical behavior of a structural element under specific constraints. Understanding this process is crucial for aspiring engineers and scientists who wish to apply mathematical models to design and analyze mechanical systems.
For Experts: This code exemplifies the power of symbolic computation in MATLAB for solving complex differential equations arising in structural dynamics and mechanical engineering. By seamlessly integrating theoretical formulations with computational algorithms, it offers a robust framework for exploring the vibrational characteristics of beams and other structural elements. The methodology outlined here can be extended to more complex systems and loading conditions, providing a foundation for advanced research and development in the field of structural analysis and design.
flexiblelink_BVP.m
The provided MATLAB code defines a function named flexiblelink_BVP that solves a boundary value problem (BVP) for a clamped-free beam subjected to an applied moment at the free end. This problem is typical in structural mechanics, illustrating the deflection of beams under certain loading conditions. The solution is achieved through MATLAB's built-in boundary value problem solver bvp4c. Let's dissect the code for both students and experts in the field.
Code Explanation Beam Properties Definition matlab Copy code L = 1; % Length of the beam in meters EI = 69e9 * (6.35e-3)^4 / 12; % Young's modulus times moment of inertia M0 = 10; % Applied moment at the free end (N*m) Here, the beam's physical properties are specified: L is the length, EI represents the flexural rigidity (product of Young's modulus and the moment of inertia of the beam's cross-section), and M0 is the moment applied at the free end of the beam. These parameters are essential for modeling the beam's behavior under load.
Differential Equation Function matlab Copy code function dydx = beamODE(x,y) dydx = [y(2); y(3); y(4); 0]; % Since EI*w'''' = 0 end This nested function beamODE represents the system of first-order ordinary differential equations (ODEs) equivalent to the fourth-order beam differential equation (Euler-Bernoulli beam theory). In this context, y is a vector where y(1) represents the deflection w, and y(2), y(3), and y(4) are the first, second, and third derivatives of w, respectively.
Boundary Conditions Function matlab Copy code function res = beamBC(ya,yb) res = [ya(1); ya(2); EIyb(3) - M0; yb(4)]; end The beamBC function specifies the boundary conditions at both ends of the beam. For the clamped end at x=0, the deflection (ya(1)) and slope (ya(2)) are zero. At the free end (x=L), the moment (EIyb(3)) equals M0, and the shear force (yb(4), related to the third derivative of deflection) is zero.
Solving the Boundary Value Problem matlab Copy code xmesh = linspace(0, L, 100); solinit = bvpinit(xmesh, [0 0 0 0]); sol = bvp4c(@beamODE, @beamBC, solinit); An initial mesh along the beam's length and an initial guess for the solution are set up. The bvp4c function is then used to solve the boundary value problem defined by beamODE and beamBC, with the initial guess solinit.
Plotting the Solution matlab Copy code x = linspace(0, L, 100); y = deval(sol, x); figure; plot(x, y(1,:), 'LineWidth', 2); Finally, the deflection of the beam is plotted along its length. The deval function evaluates the solution at points specified by x, and the first row of y (y(1,:))—representing the beam's deflection—is plotted.
flexiblelinkmodes.m
This MATLAB script calculates and visualizes the modal frequencies of a clamped-free beam with a square cross-section, made of aluminum, using the characteristic equation method. It's a great example of applying theoretical concepts from vibrations and structural dynamics in a computational environment. Let's break down the code for clarity for both students and experts in the field.
Code Breakdown Beam Properties Definition matlab Copy code L = 1; a = 6.35e-3; I = a^4 / 12; A = a^2; rho = 2700; E = 69e9; The script starts by defining the physical properties of the beam: its length (L), the dimensions of its square cross-section (a), material density (rho), and Young's modulus (E). From these, it calculates the moment of inertia (I) and the cross-sectional area (A), which are essential for further calculations.
Calculation of EI and rhoA matlab Copy code EI = E * I; rhoA = rho * A; EI (flexural rigidity) and rhoA (mass per unit length) are calculated, which are key parameters in determining the beam's vibrational characteristics.
Characteristic Equation for Clamped-Free Beam matlab Copy code charEq = @(k) cos(k*L).cosh(kL) + 1; The characteristic equation relates the wave number k to the beam's length and boundary conditions. For a clamped-free beam, the roots of this equation determine the natural frequencies of vibration.
Finding Roots Numerically matlab Copy code for n = 1:5 kGuess = (2*n-1)pi/(2L); k(n) = fsolve(@(k) charEq(k), kGuess, options); The script uses a numerical solver, fsolve, to find the first five roots of the characteristic equation. An initial guess is provided for each root based on the expected distribution of roots for a clamped-free beam.
Calculating Modal Frequencies matlab Copy code omega(n) = sqrt(EI/rhoA) * k(n)^2; f(n) = omega(n) / (2*pi); For each root k, the script calculates the angular frequency omega and the frequency in Hertz f, using the relationship between wave number, EI, rhoA, and the properties of circular motion.
Displaying Results matlab Copy code fprintf('Mode %d: k = %.4f rad/m, \omega = %.4f rad/s, f = %.4f Hz\n', n, k(n), omega(n), f(n)); The modal frequencies are displayed in a readable format, showing the wave number, angular frequency, and frequency for each mode.
Plotting Modal Frequencies matlab Copy code plot(k, f, 'o-', 'Color', 'blue', 'MarkerSize', 8, 'LineWidth', 2); Finally, a plot of modal frequencies against wave numbers provides a visual representation of the relationship between these quantities.
flexiblelinkvibration_FDM.m
This MATLAB script simulates the vibration of a clamped-free beam with an applied tip load and boundary disturbance using the Finite Difference Method (FDM). It includes the dynamic visualization of the beam's displacement over time and records this animation as a video. Let's unpack the script for a comprehensive understanding tailored to both students and experts.
Script Overview Beam Properties and Initial Setup The script begins by defining the beam's physical properties, including length, cross-sectional dimensions, material properties (density and Young's modulus), and calculates derived properties like the moment of inertia and area. These parameters are crucial for analyzing the beam's structural behavior.
Simulation Parameters Next, parameters for the spatial (nx) and temporal (nt) resolution of the simulation, as well as the total simulation time (tf), are established. These determine the granularity and duration of the simulation.
Mesh Spacing and Time Step Calculation Calculates dx and dt to define the spatial and temporal increments for the simulation, essential for the discretization in the FDM approach.
Initial Conditions Imposes an initial deflection caused by a tip payload (P) across the beam, using a simple static deflection formula. This step sets the starting point for the dynamic simulation.
Damping, Payload, and Other Parameters Defines additional parameters such as damping ratio (ksi), mass of the tip payload (Ms), and a damping coefficient (alpha) to incorporate damping effects into the simulation.
Boundary Disturbance Creates a time-varying boundary disturbance (d) function to simulate external forces acting on the system, adding complexity to the vibrational behavior.
Main Loop for FDM The core of the script, where the FDM is applied to solve the beam's vibration equation iteratively for each time step. It incorporates both the beam's physical properties and the boundary conditions to update the displacement values.
Visualization Generates a 3D mesh plot of the beam's displacement over time, providing a visual overview of the entire simulation.
Video Recording Setup Prepares a video writer object to record the dynamic simulation, specifying the format and frame rate for the output video.
Animation and Recording Iterates through time steps at a specified interval (skipFrames), updating and plotting the beam's displacement at each frame. Each plot is captured and written to the video file, creating a time-lapse animation of the beam's vibration.
Video Finalization and Notification Closes the video file and notifies the user that the animation has been saved, providing a tangible output from the simulation.
spatialmode_numeric.m
This code is designed to analyze the deflection of a beam under a given load, specifically focusing on the spatial mode of vibration and utilizing both numerical and symbolic approaches to solve for the beam's deflection profile. It is structured for MATLAB and employs several key features of the language, including function definition, numerical methods, and symbolic computation. Here's a breakdown of its components:
Beam Properties The script starts by defining physical properties of the beam, such as its length (L), cross-section dimensions (a), moment of inertia (I), Young's modulus (E), density (rho), and cross-sectional area (A). These properties are used to calculate the beam's flexural rigidity (EI) and are essential inputs for the model that predicts how the beam will behave under load. First Mode Approximation It approximates the first mode of vibration for a clamped-free beam using a predefined constant (k1). This value is used to calculate the natural frequency (omega_1) of the beam's first mode of vibration, which is crucial for understanding how the beam will deform dynamically. Boundary Value Problem (BVP) Solution The code sets up and solves a boundary value problem (BVP) to find the beam deflection under a given load. This involves defining a mesh of points along the length of the beam (xmesh), an initial guess for the solution (solinit), and then using MATLAB's bvp4c function to solve the BVP. The differential equations (beamODE) and boundary conditions (beamBC) defined in separate functions are passed as arguments to bvp4c.
beamODE: Defines the ordinary differential equation (ODE) representing the beam's behavior. beamBC: Specifies the boundary conditions at the ends of the beam, incorporating the effect of a tip payload (P). Plotting the Numerical Solution After solving the BVP, the script plots the numerical solution for the beam deflection (phi(x)) across its length. This visualization helps in understanding the physical deformation of the beam under the specified conditions. Symbolic Solution and Plotting In addition to the numerical solution, the script also includes a symbolic solution for the beam's deflection, represented by a complex mathematical expression (phiSol1). This solution is converted into a MATLAB function (phiSol1_numeric) and evaluated over a range of x values. The symbolic solution is then plotted, providing a different perspective on the beam's deflection. This approach can offer more insight into the mathematical properties of the solution and is useful for analytical purposes. Functions beamODE and beamBC beamODE: This function models the differential equation governing the beam's deflection. It takes the position along the beam (x), current state (y), and other parameters like the angular frequency (omega), density (rho), cross-sectional area (A), and flexural rigidity (EI) to compute the rate of change of the state vector. beamBC: Implements the boundary conditions for the beam, incorporating the effect of a tip payload. It ensures that the solution satisfies physical constraints at both ends of the beam. Conclusion This script is a comprehensive example of how to apply both numerical and symbolic techniques to solve engineering problems in MATLAB. It demonstrates the use of differential equations to model physical systems, the solving of boundary value problems, and the visualization of solutions to gain insights into the system's behavior.
spatialmode_symbolic.m
The continuation of the code snippet focuses on the symbolic analysis of a beam subjected to a tip load, employing MATLAB's symbolic computation capabilities. Here's a breakdown of what this segment does:
Define Symbolic Variables and Equation This part defines symbolic variables for the deflection phi(x) and the angular frequency omega. It then specifies the beam's governing differential equation in terms of these variables, incorporating the beam's flexural rigidity (EI), density (rho), and cross-sectional area (A). The equation reflects the beam's behavior under bending and is a fourth-order differential equation representing the relationship between the beam's deflection and the applied load. Boundary Conditions Symbolically defines the boundary conditions for the beam: phi(0) == 0: The deflection at the base of the beam is zero (clamped end). subs(diff(phi, x), x, 0) == 0: The slope of the beam at the base is zero, indicating no initial tilt. subs(diff(phi, x, 2), x, L) == P*L/EI: The moment at the beam's free end is proportional to the applied load (P), its lever arm (L), and inversely proportional to the beam's flexural rigidity. subs(diff(phi, x, 3), x, L) == 0: The shear force at the free end of the beam is zero. Solve the Differential Equation Attempts to solve the differential equation given the specified boundary conditions using MATLAB's dsolve function. The solution phiSol(x) represents the beam's deflection as a function of position along its length. Calculate First Mode Shape Calculates the first mode of vibration's approximated root (k1) and substitutes it into the angular frequency omega to get omega_1. This value is used to find the specific solution phiSol1 for the first mode by substituting omega_1 into phiSol. The solution phiSol1 is then converted into a MATLAB function handle (phiSol1_func) for numerical evaluation and plotting. Plotting Generates a range of x values along the length of the beam and evaluates the symbolic solution phiSol1 at these points. Plots the evaluated solution, showing the mode shape of the beam for the first approximated root. The plot visualizes how the beam's deflection varies along its length, providing insights into its bending behavior under the applied conditions. Conclusion This part of the code demonstrates the use of MATLAB's symbolic toolbox for solving and analyzing differential equations that model physical systems, specifically the bending of beams under load. By solving the beam equation symbolically and applying boundary conditions, it finds an expression for the beam's deflection and visualizes the first mode shape. This approach is valuable for understanding complex mechanical behaviors without resorting to numerical approximation methods.
spatialmode_symbolic_inverse.m
This continuation of the code snippet dives deeper into the analysis of the beam's response, particularly focusing on adjusting boundary conditions for a clamped-free beam with a tip payload, solving for mode shapes, and addressing normalization for comparison with real-world measurements.
Adjusted Boundary Conditions and Coefficients
It introduces adjusted boundary conditions to account for the clamped-free beam's response to a tip payload. The calculation uses a matrix formulation to represent these conditions and solves for coefficients. The pinv function is used to calculate the pseudo-inverse of the matrix f to solve for the coefficients. This approach allows for the determination of the beam's response under specific boundary conditions and loadings. Symbolic Mode Shape Representation The code snippet then defines a symbolic expression for the second mode shape of the beam, phi2, using a highly complex mathematical expression derived from the coefficients and boundary conditions previously established. This expression captures the detailed behavior of the beam's deflection along its length. The symbolic mode shape is then converted into a numeric function phi2_func for evaluation and plotting purposes. Normalization and Plotting The calculated mode shape is normalized using a real tip displacement value (real tip displacement) to ensure the model's predictions align with observed physical phenomena. This normalization factor (cf) scales the computed deflection to match real-world measurements. The script plots the normalized second mode shape across the beam's length, providing a visual representation of how the beam deflects under the given conditions. Additional Calculations The script also performs calculations to assess the beam's curvature (D2phi2a) and deflection (phi2a, phi2L) at specific points along its length, particularly at the midpoint and the tip. These values can provide further insights into the beam's behavior and structural integrity under load. This advanced segment of the code demonstrates a comprehensive approach to modeling and analyzing the deflection of a clamped-free beam under a tip load using MATLAB's symbolic computation capabilities. By adjusting boundary conditions, solving for mode shapes, and employing normalization, the analysis provides a deep understanding of the beam's structural response. This method can be particularly useful in fields such as mechanical engineering, structural analysis, and materials science, where accurate modeling of physical systems is crucial for design and analysis.
flexiblelink_lagrange.m
This segment of the code advances into the realm of dynamic systems and control, focusing on a system that might involve a motor (represented by the angle q1(t)) and the deformation of a beam in its second mode (q2(t)). It leverages MATLAB's Symbolic Math Toolbox to derive the equations of motion using the Euler-Lagrange equation, a fundamental principle in the dynamics of mechanical systems. Here's a breakdown: Symbolic Variables and Functions The script defines symbolic functions representing the motor angle and the deformation of the beam in its second mode, respectively. Additionally, symbolic constants such as the inertias (Ih,Ib), density (ρ), angular frequency of the second mode (ω2), a normalization factor (γ2), and the torque (τ) are declared. Derivatives and Lagrangian Formulation Derivatives with respect to time (t) are computed, necessary for formulating the Lagrangian (L) of the system. The Lagrangian is defined as the difference between the kinetic and potential energies of the system, incorporating both the motor and the beam's second mode dynamics. Euler-Lagrange Equations The Euler-Lagrange (EL) equations are derived from the Lagrangian for both q1 and q2, representing the fundamental equations of motion for the system. These equations are obtained by taking the derivative of the Lagrangian with respect to the function and its time derivative, then setting the difference of these derivatives equal to the external force/torque applied to the system (τ in this case). For q1: The EL equation considers the inertia of the motor and beam and the coupling between the motor's angular motion and the beam's deformation. For q2: It focuses on the dynamics of the beam's deformation, considering its natural frequency and the effect of coupling with the motor's motion. This approach demonstrates a sophisticated method for modeling the dynamics of a coupled system consisting of a motor and a beam in vibration. By utilizing the symbolic computation capabilities of MATLAB, the script efficiently derives the equations of motion that govern the system's behavior. These equations are essential for understanding the system's dynamics and can be used for further analysis, such as stability assessment, control strategy development, or simulation of the system's response to external inputs. In practical applications, these derived equations of motion could inform the design and control of mechanical systems where precise movement and vibration control are critical, such as in robotics, aerospace structures, or precision manufacturing equipment.
flexiblelinksim.m
The provided code is a MATLAB script designed for simulating and analyzing the dynamic behavior of a system represented by two linked variables, q1 and q2, under certain conditions. It employs ordinary differential equations (ODEs) to model the system's motion and uses MATLAB's ODE solver ode45 for numerical integration. Here's a breakdown of its components and functionality:
Constants I_h: Moment of inertia, a physical quantity expressing an object's tendency to resist angular acceleration. rho: Density or a specific parameter related to the system that influences its dynamics. omega: Natural frequency of q2, indicating how q2 oscillates in the absence of damping or external forces. gamma: Damping factor, a coefficient that describes the rate at which the system's motion is attenuated due to resistive forces. Initial Conditions q0: A vector specifying initial conditions for q1, q1dot (velocity of q1), q2, and q2dot (velocity of q2). It sets the starting point of the simulation. Time Span tspan: Defines the duration of the simulation, from 0 to 1 second. Solver Configuration An options set for the ODE solver (odeset) is defined to control the maximum step size during numerical integration, enhancing stability. ODE Solver The ode45 function is used to solve the system of differential equations defined in the stateSpace function over the specified time span. It computes the system's states (x) at various time points (t) based on the initial conditions and the equations governing the system's dynamics. Plotting The script generates a series of plots to visualize the positions (q1, q2) and velocities (q1dot, q2dot) of the system components over time, facilitating analysis of their behavior under the simulated conditions. stateSpace Function This function calculates the derivatives of the system's state variables (dxdt) at any given time (t), based on the current state (x), system constants (I_h, rho, omega, gamma), and a control input (tau). tau is modeled as a smoothly transitioning function using a sigmoid curve, which represents an external force or torque applied to the system. The equations for q1ddot and q2ddot represent the accelerations of q1 and q2, respectively, derived from the system's dynamics and the control input.
In summary, this MATLAB script is a tool for simulating the dynamics of a system with two degrees of freedom, taking into account factors like inertia, damping, and external control inputs. It provides insights into how the system responds over time to a smoothly varying external force, with applications potentially in fields such as mechanical engineering, robotics, or any area involving the dynamic analysis of linked systems.
flexiblelinksimulation.slx is the Simulink version, load the flexiblelinkparams.m file before run the simulation.
rotatingflexiblebeammovingmass_symbolic.m
The provided MATLAB code snippet is an advanced example of applying symbolic computation for deriving the equations of motion of a mechanical system, specifically optimized for a beam and a particle system. This snippet demonstrates the use of MATLAB's Symbolic Math Toolbox for precomputing, optimizing mathematical expressions, and efficiently deriving equations of motion using the Lagrangian dynamics approach. Here's a detailed explanation:
Variable Definitions Symbolic variables (syms) for the system's states and parameters are defined, including positions, velocities, accelerations, geometrical and material properties, and external forces. Precomputation for Optimization Expressions involving the ratio of a variable x and u to the beam length L are precomputed to avoid redundant calculations during the derivation of the system's dynamics. This step significantly improves computational efficiency. Shape Functions for Beam Element Optimized shape functions N1 and N2 are defined for the beam element based on the beam's geometry. These functions are critical for finite element analysis, representing how different points of the beam deform under loads. Stiffness and Mass Matrices The stiffness matrix Kf of the beam is defined, which relates to the beam's resistance to deformation. It's crucial for understanding the beam's structural behavior. The elastic mass matrix Mf and mass coupling term Mrf are computed. These matrices represent the distribution of mass within the beam and its influence on the system's dynamics. Kinetic and Potential Energy The kinetic energy T and potential energy V of the system are calculated, incorporating contributions from both the beam and the particle. These energies are essential for formulating the system's Lagrangian, which is the difference between the kinetic and potential energies. Lagrangian Dynamics The Lagrangian Lag is formulated, and from it, the equations of motion are derived using the principle of least action. This involves taking derivatives of the Lagrangian with respect to the system's state variables and their derivatives, then simplifying the resulting expressions. Equations of Motion The script efficiently groups similar derivative operations to optimize the derivation of the equations of motion. It calculates the left-hand side (LHS) of the equations of motion for each state variable (q1, q2, u, th), which represent the dynamics of the system. Display Finally, the optimized equations of motion for each variable are displayed. These equations are essential for understanding how the system behaves over time, especially under various external forces and moments. This code is a sophisticated example of utilizing symbolic computation for mechanical system analysis. It highlights the power of MATLAB for handling complex mathematical operations, optimizing calculations, and providing insights into the dynamics of mechanical systems, which can be invaluable in fields such as structural engineering, robotics, and physics.
acrobot_zerodynamics.m
The provided MATLAB script is a comprehensive analysis tool for the dynamics and zero-dynamics analysis of an acrobot system. An acrobot is a planar robot with two links connected by an actuated joint, resembling a simplified version of a gymnast swinging on a high bar. This script employs symbolic mathematics, dynamic system simulation, and linearization techniques to model, simulate, and analyze the system's behavior. Here's a breakdown of the key components and steps in the script:
Symbolic Variable Definition The script starts by defining symbolic variables for the angles (theta1, theta2), torque (tau), mass (m1, m2), length (l1, l2) of the links, and gravitational acceleration (g). These variables are used to describe the physical properties and states of the acrobot system. Derivatives It calculates the derivatives of theta1 and theta2 with respect to time (t), which are needed for dynamic analysis. Kinetic and Potential Energy The kinetic (T) and potential (V) energies of the system are defined using the physical parameters and state variables. These energies are crucial for formulating the Lagrangian, which represents the difference between kinetic and potential energy. Lagrangian Mechanics The script employs Lagrangian mechanics to derive the equations of motion. This involves the calculation of the Lagrangian (L = T - V) and applying the Euler-Lagrange equations to find the dynamic equations. However, the actual execution of deriving equations of motion using Euler-Lagrange equations is commented out, indicating it might have been done manually or in another part of the script. System Parameters Specific values for the mass, length of links, and gravitational acceleration are assigned, transitioning from symbolic representation to a numerical setup ready for simulation. Equations of Motion The equations of motion are explicitly defined using the system parameters, allowing for the calculation of the second derivatives of theta1 and theta2 (ddth1, ddth2), which represent angular accelerations. Solving for Angular Accelerations The script solves the equations of motion for ddth1 and ddth2 given a torque input (tau), enabling the analysis of how the system responds to external forces. Defining the ODE Function for Dynamics Simulation A function handle (odeFun) for the system's differential equations is created, integrating the defined dynamics with respect to a step input for torque. This function is suitable for numerical ODE solvers in MATLAB. Simulation The script simulates the full system dynamics using ode45, a MATLAB function for solving ordinary differential equations, with specified initial conditions and a time span. Plotting It visualizes the dynamics of theta1 and theta2 over time, offering insights into the system's behavior under the defined conditions. Linearization The system is linearized around the equilibrium point (assumed to be the upright position), which simplifies the analysis of system stability and control design. This involves calculating Jacobian matrices A and B for the system dynamics and input, respectively. State-Space Model and Simulation A state-space model is created from the linearized system, and the response to a step input in torque is simulated and plotted. This analysis is crucial for understanding the system's behavior in a linearized context, which is easier to control and analyze compared to the nonlinear dynamics. Overall, this MATLAB script is a thorough tool for studying the dynamics of an acrobot system, from symbolic derivation of equations of motion to numerical simulation and linearization for control purposes. It exemplifies a systematic approach to modeling, simulation, and analysis that is critical in robotics and control engineering.
For the Simulink simulation 'acrobot_zerodynamics_simulink_control.slx', first run the 'acrobot_parameters.m'file.