【Advanced】Robot Dynamics Simulation and Control Based on MATLAB

# 2.1 Establishing Equations of Robot Dynamics

Robot dynamics equations describe the relationship between robot motion and the torques acting upon it. There are two primary methods for establishing these equations: the Lagrange equations and the Newton-Euler equations.

### 2.1.1 Lagrange Equations

Lagrange equations are a type of dynamic equation based on the principle of least action. For an n-degree-of-freedom robot, the Lagrange equation is:

d/dt (dL/d(dq/dt)) - dL/dq = Q

Where:

* L is the Lagrangian function, equal to the kinetic energy minus the potential energy of the robot
* q is the vector of generalized coordinates
* Q is the vector of generalized forces

### 2.1.2 Newton-Euler Equations

Newton-Euler equations are a type of dynamic equation based on Newton's second law and Euler angles. For a rigid robot, the Newton-Euler equation is:

F = ma
M = Iα + ω × (Iω)

Where:

* F is the resultant external force acting on the rigid body
* M is the resultant external moment acting on the rigid body
* a is the acceleration of the rigid body
* α is the angular acceleration of the rigid body
* I is the inertia tensor of the rigid body
* ω is the angular velocity of the rigid body

# 2. MATLAB Robot Dynamics Simulation Techniques

MATLAB, as a powerful numerical computing and visualization tool, has wide applications in the field of robot dynamics simulation. This section will introduce related techniques in MATLAB for robot dynamics simulations, including the creation of robot models, kinematic and dynamic analysis, and the analysis and visualization of simulation results.

### 2.1 Establishing Equations of Robot Dynamics

Robot dynamics equations describe the relationship between robot motion and the torques and inertias acting upon it. Establishing robot dynamics equations is the basis for robot dynamics simulation and control.

#### 2.1.1 Lagrange Equations

Lagrange equations are a method for establishing dynamic equations based on the principle of least action. The general form of the Lagrange equation is:

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = Q_i$

Where:

- $L$ is the Lagrangian, representing the kinetic energy minus the potential energy of the system
- $q_i$ is the generalized coordinate
- $\dot{q}_i$ is the generalized velocity
- $Q_i$ is the generalized force

For a robot system, the Lagrangian can be expressed as:

$L = K - P$

Where:

- $K$ is the kinetic energy
- $P$ is the potential energy

#### 2.1.2 Newton-Euler Equations

Newton-Euler equations are a method for establishing dynamic equations based on Newton's second law and Euler angles. The general form of the Newton-Euler equation is:

$F_i = m_i a_i$

Where:

- $F_i$ is the resultant external force acting on the $i$th rigid body
- $m_i$ is the mass of the $i$th rigid body
- $a_i$ is the acceleration of the $i$th rigid body

For a robot system, the Newton-Euler equation can be expressed as:

$H(q)\ddot{q} + C(q,\dot{q})\dot{q} + G(q) = \tau$

Where:

- $H(q)$ is the inertia matrix
- $C(q,\dot{q})$ is the centrifugal and Coriolis force matrix
- $G(q)$ is the gravity matrix
- $\tau$ is the joint torque

### 2.2 Robot Dynamics Simulat***

***monly used functions and toolboxes include:

- Robotics System Toolbox
- Simulink
- Simscape Multibody

#### 2.2.1 Creating Robot Models

Creating a robot model in MATLAB requires specifying the robot's geometric parameters, inertia parameters, and joint types. The `robot` function in the Robotics System Toolbox can be used to create robot models.

robot = robot('model.urdf');

Where: `model.urdf` is the robot model file in URDF format.

#### 2.2.2 Kinematic and Dynamic Analysis

MATLAB provides various functions for performing kinematic and dynamic analysis of robots, including:

- `forwardKinematics`: Forward kinematics analysis
- `inverseKinematics`: Inverse kinematics analysis
- `jacobian`: Jacobian matrix calculation
- `inertia`: Inertia matrix calculation
- `dynamics`: Dynamic equation solving

% Forward kinematics analysis
T = forwardKinematics(robot, q);

% Inverse kinematics analysis
q = inverseKinematics(robot, T);

% Jacobian matrix calculation
J = jacobian(robot, q);

% Inertia matrix calculation
H = inertia(robot, q);

% Dynamic equation solving
tau = dynamics(robot, q, dq, ddq);

### 2.3 Analysis and Visualization of Simulation Results

M***monly used visualization tools include:

- `plot`: Plotting line graphs
- `surf`: Plotting surface graphs
- `mesh`: Plotting mesh graphs
- `animation`: Creating animations

% Plotting joint angle curves
plot(time, q);

% Plotting joint torque curves
plot(time, tau);

% Plotting robot motion trajectories
figure;
hold on;
for i = 1:length(time)
    T = forwardKinematics(robot, q(:,i));
    plot3(T(1,4), T(2,4), T(3,4), 'ro');
end
hold off;

# 3. MATLAB Robot Control Algorithms

### 3.1 PID Control

#### 3.1.1 Design and Tuning of PID Controllers

PID (Proportional-Integral-Derivative) controllers are widely used classical control algorithms in robot control. The design process includes determining controller parameters (proportional gain Kp, integral gain Ki, and derivative gain Kd).

**Proportional Gain (Kp)**: Kp determines the response speed of the controller; increasing Kp can enhance the response speed, but too high values can lead to system instability.

**Integral Gain (Ki)**: Ki eliminates steady-state errors; increasing Ki can reduce steady-state errors, but excessive values can cause system oscillations.

**Derivative Gain (Kd)**: Kd predicts future system behavior; increasing Kd can improve system stability, but too high values can make the system sensitive to noise.

There are various methods for tuning PID parameters, including:

- **Ziegler-Nichols Method**: Based on system step response curves, initial values for Kp, Ki, and Kd are determined.
- **Trial-and-Error Method**: Systematically adjusting parameters to f