(TIỂU LUẬN) REPORT TOPIC MODELING AND SIMULATING THE THREEWHEELED MOBILE ROBOT

Differential Drive A different wheeled robot is a mobile robot is a mobile robot whose movement is based on separately driven wheels placed on their side of the robot body.. Let the angl

Introduction

Wheeled Mobile Robot

Wheeled Mobile Robots (WMR) require precise torque application to their wheels to achieve desired platform movement, making dynamic control essential Effective motion control algorithms must consider the system's dynamic properties to ensure accurate navigation Typically, cascade control schemes are employed, featuring an outer velocity controller and an inner torque or motor current controller to optimize performance and stability.

The outer controller sets the necessary system velocities to help the system reach the reference pose or follow a designated trajectory Meanwhile, the inner controller operates at a faster rate, computing the precise torques needed to achieve the velocities specified by the outer controller This hierarchical control structure ensures accurate and efficient navigation of the system along planned paths.

Some typed of Wheeled Mobile Robot

A different wheeled robot is a mobile robot is a mobile robot whose movement is based on separately driven wheels placed on their side of the robot body.

Robots with differential drive can change direction by adjusting the relative rotational speeds of their wheels, eliminating the need for a separate steering mechanism To maintain stability and prevent tilting, these robots often incorporate one or more caster wheels This design allows for smooth and precise maneuverability, making it ideal for various robotic applications.

Bicyce Drive also have 2 wheels as

Differential Drive, but its wheels are arrange in a straight line, and usually only one wheel is active and other can control the steering angle, similar like a bicycle.

Tricycle Drive is a combination of WMR mentioned above, it has 3 wheels:

The vehicle features two rear wheels arranged coaxially, with a single front wheel used for steering Two of the three wheels are connected to an actuator to precisely control speed and direction, while the remaining wheel operates freely This configuration enhances maneuverability and enables accurate control of both speed and steering angles.

WMR has a structure similar to a car with 2 front wheels that can change the steering angle.

An Omni Directional Wheel is a versatile wheel designed to freely roll in multiple directions, enhancing mobility and maneuverability Besides the commonly known types of Wheel Motor Robots (WMR) discussed earlier, there are various other WMR types, with these examples representing some of the most typical and widely used designs in the industry.

Choosing WMR three wheels

In this topic, We choose a Tricycle Drive with 2 rear wheels as active wheels attached with actuators to control and front wheel as free wheel.

Modeling the Moblie Robot Three Wheels

Kinetic Model

WMR is illustrates as in the figure H2.1:

ICR: instantaneous center of rotation of vehicle R(t): instantaneous radius of vehicle’s trajectory

 : angular speed of vehicle around the center of ICR v : longitudinal vehicle velocity r : wheel’s radius

 : the vehicle’s orientation – angle between WMR and O m X m axis

The model is placed in a general coordinate system  X g ,Y g  , and the coordinate system of motion associated with WMR X m ,Y m  The state vector of the vehicle in the general coordinate system is:

Because the movement of the vehicle through transmission of 2 rear wheels that moniting the WMR direction; therefore, the vehicle's longitudinal velocity is determined by:

The equation of external kinematic of WMR on the general coordinate system is determined as follows:

In order to eleminate the complex in presentation, we temporarily ignore the dependence of the quantity into time, the above equation can be rewritten as:

Or we can write as following matrix:

In which the directions of motion are:

System equation 2.5 is rewritten as follows:

Beside, the WMR can only move along the wheels, not drift Therefore, the motion of the WMR is constrained:

The relationship between (x2,y2) with (x,y) as following:

From 2.11, we have constrain matrix A as following:

Dynamic Model

Model 3-wheel mobile robot is considered within 2 rear wheels as active driven by two engines, font wheel as free one These actuators will generate torques

The dynamic model of WMR’s motion is described by the Lagrange formula as follows:

: the difference between the kinetic and potential energy of the system

P : the power lost due to friction k : the index of the general coordinate component q k

8 g k : gravity analysis in the direction of q k

 d : disturbance component in direction of q k

         k : component Lagrange associated wih he jth constrain according to q k a jk : constrain cefficient according to q k

The dynamic model 2.13 can be rewritten in matrix as follows:

M  q  : positive definte matrix of mass and inertia

V  q,q  :vector of Coriolis force and centrifugal force

E  q  : matrix tránition form actuation space to coordinate space u : input torque vector

In order to simplify 2.16 , we temporarily ignore the dependence on q Then 2.16 becomes

Beside, taking derivative both sides od 2.7 we have : q  S.v  S.v Then

Phần 2: Mô hình hóa đối tượng Mobile Robot 3 bánh

Multiplying both sides of 2.18 with S T we have:

From 2.7 and 2.21, we can describe the state space

Consider for the dynamic modeil, within m, J are mass anf torque around ICR respectively; and ignore disturbance component in formula 2.15

Meanwhile, we consider that WMR just run on even surface That means its potential energy is constant and q k component disappeared in equation 2.15.

The WMR’s kinetic energy is:

The Lagrangian function is calculated by:

In the ideal condition, ignoring the friction that means lost power equal 0 (P=0).

In order to 2 rear wheels can run with longitudinal velocity

The combined force to pull the WMR moving is ; considering in two components x and y we have:

The coordinate system of motion associated with WMR (Wheeled Mobile Robots) involves traction forces exerted by the two wheels, which generate a torque that causes the vehicle to rotate about its axis This torque, denoted as M, determines the rotational behavior of the vehicle and is critical for understanding its control and maneuverability.

( R  L ) 2r Subtituting 2.27, 2.28 and 2.29 into 2.26 we have:

Knematic model 2.30 is writen in the matrix as follow:

From that, state space 2.22 can be represented as following:

Because of the non-linear of system, in order to design controller, we normally separated model into 2 parts, including:

Where matrixs M,E, V are defined by 2.32

Methods Wheel Mobile Robot

Control Overview

To facilitate control design, the vehicle's state model is commonly divided into two components: kinematics and dynamics The development of control strategies is then based on these two models, enabling more precise and effective vehicle management.

To design an effective controller for a WMR, a traditional cascade control scheme with two control loops is employed, ensuring precise and stable operation The outer loop, known as the speed control or kinematic control loop, calculates the desired speed to ensure accurate movement The inner loop, the torque or dynamic control loop, manages the motor torque to achieve the speed set by the outer loop The outer loop's kinematic controller provides the target speed value to the inner loop, which then adjusts the motor torque accordingly This hierarchical control structure enhances system responsiveness and robustness A detailed block diagram illustrates how these control loops are interconnected to optimize WMR performance Incorporating this classic control strategy aligns with SEO best practices by emphasizing terms like "WMR control," "cascade control scheme," and "kinematic and dynamic control loops," making the content highly relevant and identifiable for search engines.

Beside, the above control system can also be represented in another way according to the EMR method as follows:

Basic approaches

Let the angle of direction at time t is φ(t) và and desired angle is φref(t), the control angle deviation :

In this case, the variable to be controlled is φ(t) and in order to control φ(t) to reach the desired value we need to control the bias ( ) to zero.

The direction angle of the robot is expressed through the system of differential equations of the direction angle as follows:

In this case, the control variable is selected as    t , it is determined corresponding to the angular deviation ( ) through proportional controller as follow:

This equation is approximated to:

To effectively follow a desired trajectory or maintain a specific state, translational control is essential, which involves managing the vehicle's longitudinal movement The most suitable approach is implementing a control system that adjusts proportionally to the distance between the vehicle's current position and the reference position This proportional control ensures accurate and smooth trajectory tracking, enhancing overall vehicle performance.

This section combines angular and translational control methods to effectively guide the robot to a reference state One key approach is controlling the robot to a specific set point, ensuring precise positioning and stability By integrating these control strategies, we can achieve accurate and reliable movement toward designated targets in practical applications.

In this scenario, the WMR is required to reach the reference position without needing the final direction angle The robot's direction is continuously controlled to follow the path toward the reference location, with the direction from the robot's current position to the reference point denoted as φr This approach ensures precise navigation by dynamically adjusting the robot's orientation based on the reference position, optimizing movement accuracy without reliance on a predetermined final heading.

Vận tốc dài và góc lái của bánh trước được điều khiển như sau:

The longitudinal velocity and steering direction of the front wheel are controlled as following:

The existing control law faces challenges, such as the robot potentially overshooting the reference point due to consistently positive speed, leading to continued acceleration as the distance increases Additionally, passing through the reference point can cause abrupt reversals in the reference direction angle, disrupting smooth navigation To address these issues, it is essential to modify the velocity and steering angle formulas, including a control strategy that guides the robot toward a reference pose via an intermediate point for more stable and accurate path tracking.

In this scenario, the robot must not only reach a specified position but also align to a given orientation, which is straightforward by applying the previously mentioned control rules The core idea of this method involves using an intermediate point, (x_t, y_t), positioned at a distance r from the reference point, ensuring that the direction from the intermediate point to the reference matches the desired orientation This approach simplifies the process of achieving both position and direction accuracy in robotic navigation.

The idea of that method is to use an intermediate point  x t , y t   placed a distance r from the reference point

Thuật toán điều khiển robot gồm hai giai đoạn chính: Giai đoạn đầu, robot sẽ được điều khiển để di chuyển đến điểm trung gian Khi khoảng cách từ vị trí hiện tại của robot đến điểm trung gian đủ nhỏ, robot sẽ chuyển sang giai đoạn tiếp theo để hoàn tất quá trình điều khiển.

2, điều khiển robot về điểm tham chiếu.

The control algorithm is divided into two stages to enhance navigation accuracy In the first stage, the robot is guided toward an intermediate point, with the transition to the second stage occurring once the distance between the robot's position and the reference point falls below a specified threshold In the second stage, the robot is precisely driven to the reference point, ensuring smooth and efficient movement Additionally, the system manages the transition from the reference state to following the reference trajectory for optimal path tracking.

The trajectory is segmented into multiple small sections, with reference states serving as the key waypoints A reference orbit comprises a sequence of reference states, denoted as Ti, where i equals 1, 2, 3, and so on up to n The robot is designed to move sequentially through these reference states, progressing from T1 to T2, and continuing until reaching Tn, ensuring precise and structured navigation.

Orbit Following Control

3.3.1 Following the trajectory using basic approaches.

Quỹ đạo tham chiếu được hiểu như một vị trí chuyển động tham chiếu để xác định các điểm trong quá trình trích mẫu Tại mỗi thời điểm, điểm tham chiếu được định nghĩa dựa trên vị trí hiện tại của quỹ đạo tham chiếu theo thời gian, với các tọa độ như x_ref(t) và y_ref(t) Điều này giúp duy trì độ chính xác và nhất quán trong quá trình xử lý dữ liệu, đồng thời đảm bảo hệ thống hoạt động ổn định theo yêu cầu của các phép đo.

 t    Việc điều khiển tới điểm tham chiếu này được áp dụng luật điều khiển như công thức (3.2).

Similar to section 3.2.2.c, the reference trajectory can be considered as the set of reference movement positions, which are updated at each sampling time The reference point is recalculated based on the current position of the reference orbit at time t, with coordinates (x_ref(t), y_ref(t)), ensuring accurate trajectory tracking and control.

3.3.2 Analysis feedforward and feedback elements

The method outlined in section 3.3.1 is simple to implement; however, it is highly sensitive to disturbances within the control loops To address this vulnerability, incorporating a feedforward disturbance compensation component is essential for maintaining system stability and performance.

A planar differential system is characterized by the presence of flat outputs, allowing all system states and inputs to be expressed as functions of these flat outputs and their finite derivatives This property enables the direct computation of system variables from flat outputs without the need for integration As a result, given a reference trajectory, control inputs can be efficiently determined solely based on the flat outputs and their derivatives, simplifying system analysis and control design.

We can easily see that the 3 wheeled mobile robot object in question is a planar differential system Indeed:

Formula (3.3) provides the method for calculating reference inputs aligned with a planned trajectory, serving as an open-loop controller for the robot's kinematic model This approach ensures precise following of the set trajectory, making it essential for effective motion planning and control.

The core concept of this method involves applying transformations to linearize the system's input, enabling a linear relationship between the new input and the output This approach simplifies the design of linear controllers for the system The feedback linearization process begins with selecting appropriate flat outputs, ensuring the number of outputs matches the number of inputs for optimal system control.

- Selecting the appropriate flat outputs, the number of outputs should be equal to the number of inputs

- Differentiate the outputs and check for the occurrence of the inputs, repeating until all inputs appear.

- The system of equations is solved for the highest derivative of each input.

The 3-wheel mobile robot operates as a planar differential system, with its flat outputs being the position coordinates x(t) and y(t) The system's control relies on these flat outputs, with the first derivatives—representing the velocity components—serving as key inputs for trajectory planning and control Understanding this structure is essential for effective navigation and precise maneuvering of the robot in two-dimensional space.

The equation has sufficient input vs and The equation is rewriten as follows:

So with this transformation, the input of the system becomes

 u 1 u 2  T    x y  T and state modelz   x x y y  T is described as :

Then weed need to design the controller for this new linear system The reference trajectory is given by , then the reference trajectory can be determined as : We have:

The difference between the actual system state and the reference state is z  z  z ref Then:

Using the pole setting method, we define the controller K:

3.3.4 Development of tracking kinetic trajectory.

The position deviation of the robot in general coordiante system with the reference position is:

The position deviation of the robot in the coordinate system of motion associated with WMR is:

Where v ref and w ref are linear reference velocity: v fb and w fb are signals that are determined depending on the control law we choose The error equation is rewritten

Realize zero-error ( e x = e y = e  =0) is an equilibrium when both components respond

In section 3.3.4, we linearize the nonlinear error model around the zero-error point to improve accuracy This approach ensures that deviations in steering direction and steering angle are effectively corrected through the feedback velocity (v_fb) and angular velocity (ω_fb), enhancing the control precision of the vehicle.

The control coefficient k x , k y , k  are chosen so that the poles of the system are in the appropriate position in the s domain.The system has 3 poles, in which 1 real pole and

2 conjugate complex poles The poles are placed in a fixed position with Based on the characteristics polynimial of the closed loop system:

We have the control coefficient:

Phần 5: Mô phỏng hệ thống điều khiển trên Matlab Simulink

In reality the coefficients in 3.41 are not easy to apply, cause ky(t) becomes quite large when v ref is small.

We draw two important observations:

The controller design relies on a linear model, which accurately represents system behavior near specific points like zero-error conditions However, this linear approximation becomes less effective when the error significantly deviates from these points Consequently, the system's performance may degrade when operating outside the valid range of the linear model, highlighting the importance of understanding its limitations in control applications.

Although the system is linear, it remains time-dependent, which affects its behavior over time The stability of the system is primarily determined by the placement of its poles in the s-domain; specifically, if all poles are located in the left half-plane, the system will be stable Conversely, if any pole exists in the right half-plane, the system becomes unstable Understanding pole placement is crucial for analyzing and designing stable, time-dependent linear systems.

Even though, the linear control law is still commonly used because it is simple, easy to adjust in practice, and the performance is acceptable.

Part 4 : Designing Controller For Three

Based on the content presented in part 3, we design an outer loop controller for Mobile robot based on the candidate function Lyapunov.

Thus, the reference velocity of the robot when moving on the reference trjectory is:

The position deviation on the coordinate system of motion associated with WMR is:

Where matrix C is a positive define 2x2 control matrixThen:

Control System On Matlab Simulink

Knematic Model

Dynamic Model

5.3 Bộ điều khiển vòng ngoài Kynematic Controller

Getting the initial position of WMR is [x(0);y(0);  (0)]= [2; 2; pi/4]

In other hands, the reference trajectory:

The result of robot’s trajectory:

At this time, the initial values stay the same as case one but increasing the frequency 6 times:

This situation changing matrix C becomes:

We get the result as following:

The matrix C undergoes significant changes, leading to slight variations in the robot's trajectory Despite these differences, the robot's adaptive capabilities with the dynamic controller remain highly effective, demonstrating robustness when integrated with the kinematic controller This synergy minimally impacts the overall system performance, highlighting the controller's efficiency in adapting to dynamic changes.

In final situation, we change the values     ang g= 300.

Thus, we get the result:

Simulink Result