Robot Control

Introduction

Robot control is a complex system that integrates mechanical engineering and electronic control. This article introduces the fundamental theory of robot control along with practical implementation examples.

Fundamental Control Theory

The basis of robot arm control lies in position and velocity control, for which PID controllers are commonly utilized.

Mathematical Basis of PID Control

The output of a PID controller is expressed by the following equation:

$u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}$

Where:

• $u(t)$: Control signal
• $e(t) = r(t) - y(t)$: Error (setpoint - actual value)
• $K_p, K_i, K_d$: Proportional, integral, and derivative gains, respectively

Sample Implementation

Below is a simple implementation of a PID controller in Python:

class PIDController:
    def __init__(self, kp, ki, kd):
        self.kp = kp
        self.ki = ki
        self.kd = kd
        self.previous_error = 0
        self.integral = 0
    
    def update(self, setpoint, measured_value, dt):
        error = setpoint - measured_value
        self.integral += error * dt
        derivative = (error - self.previous_error) / dt
        
        output = (self.kp * error + 
                  self.ki * self.integral + 
                  self.kd * derivative)
        
        self.previous_error = error
        return output

# Usage example
controller = PIDController(kp=1.0, ki=0.1, kd=0.5)
setpoint = 100.0  # Target value
measured_value = 50.0  # Current value
control_signal = controller.update(setpoint, measured_value, dt=0.01)

Inverse Kinematics

To control the end-effector position of a robot arm, it is necessary to calculate the angles of each joint.

Inverse Kinematics for a 2D Arm

For a two-link arm with link lengths $l_1$ and $l_2$, to move the end-effector to position $(x, y)$, the joint angles $\theta_1$ and $\theta_2$ are calculated as follows:

$\theta_2 = \arccos\left(\frac{x^2 + y^2 - l_1^2 - l_2^2}{2l_1l_2}\right)$

$\theta_1 = \arctan2(y, x) - \arctan2(l_2\sin\theta_2, l_1 + l_2\cos\theta_2)$

Robot State Management in JavaScript

class RobotArm {
    constructor(linkLengths) {
        this.linkLengths = linkLengths;
        this.jointAngles = new Array(linkLengths.length).fill(0);
    }
    
    forward_kinematics() {
        let x = 0, y = 0;
        let currentAngle = 0;
        
        for (let i = 0; i < this.linkLengths.length; i++) {
            currentAngle += this.jointAngles[i];
            x += this.linkLengths[i] * Math.cos(currentAngle);
            y += this.linkLengths[i] * Math.sin(currentAngle);
        }
        
        return { x, y };
    }
    
    move_to(targetX, targetY) {
        // Simple inverse kinematics using gradient descent
        const learning_rate = 0.01;
        for (let iter = 0; iter < 1000; iter++) {
            const { x, y } = this.forward_kinematics();
            const error = Math.sqrt((x - targetX)**2 + (y - targetY)**2);
            
            if (error < 0.01) break;
            
            for (let i = 0; i < this.jointAngles.length; i++) {
                this.jointAngles[i] += learning_rate * error;
            }
        }
    }
}

Dynamic Modeling

Robot arm dynamics are described by the Lagrangian equations:

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

Where $L = T - V$ (kinetic energy - potential energy), and $\tau_i$ represents the joint torque.

Implementation Considerations

When implementing robot control, attention should be paid to the following points:

1. Sampling Time: Set the control loop cycle to be sufficiently short (typically 1ms or less).
2. Anti-Windup: Prevent the integral term from increasing indefinitely.
3. Sensor Noise: High derivative gains can amplify signal noise.
4. Mechanical Constraints: Adhere to joint range of motion limits as well as maximum speed and torque specifications.

Conclusion

Understanding robot control requires balancing both theory and implementation. It starts with PID control and evolves toward more complex nonlinear control strategies.