3.2 PID control and trajectory tracking

PID control is a crucial technique in robotics for precise system management. It combines proportional, integral, and derivative components to minimize errors in various applications, from motor speed control to humanoid robot balance.

Designing PID controllers for robots involves system modeling, tuning methods, and performance criteria. Gain selection follows a systematic approach, starting with proportional control and adding integral and derivative components as needed. Simulation and testing validate the design's effectiveness.

PID Control Fundamentals

Principles of PID control

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• PID control components work together to minimize error in control systems
  • Proportional (P) responds to current error with gain $K_p$
  • Integral (I) accumulates past errors with gain $K_i$
  • Derivative (D) anticipates future errors with gain $K_d$
• PID control equation combines components: $u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}$
• Error calculation measures deviation from desired state: $e(t) = \text{[setpoint](https://www.fiveableKeyTerm:Setpoint)} - \text{measured value}$
• Applications in robotics enhance precision and stability
  • Motor speed control maintains consistent RPM
  • Joint position control ensures accurate limb placement
  • Balance control for humanoid robots prevents falling
  • Autonomous vehicle steering keeps vehicles on course

PID controller design for robotics

• System modeling captures dynamic behavior
  • Transfer function representation describes input-output relationships
  • State-space model provides internal system dynamics
• Tuning methods optimize controller performance
  • Ziegler-Nichols method uses oscillation characteristics
  • Cohen-Coon method suited for processes with time delay
  • Manual tuning allows fine-grained adjustments
• Performance criteria guide design process
  • Rise time measures initial response speed
  • Settling time indicates stabilization period
  • Overshoot quantifies maximum deviation
  • Steady-state error evaluates long-term accuracy
• Gain selection process follows systematic approach
  1. Start with P control for basic responsiveness
  2. Add I control to eliminate persistent errors
  3. Introduce D control to improve transient behavior
• Simulation and testing validate design
  • Software tools (MATLAB, Simulink) enable virtual prototyping
  • Hardware-in-the-loop testing bridges simulation and real-world performance

Trajectory Tracking and System Analysis

Trajectory tracking in manipulators

• Path planning determines optimal route
  • Joint space vs. Cartesian space planning affects smoothness
  • Polynomial trajectories ensure continuous motion
  • Spline interpolation creates smooth curves between waypoints
• Inverse kinematics calculates required joint angles
  • Analytical methods provide closed-form solutions
  • Numerical methods (Newton-Raphson) handle complex geometries
• Feedforward control anticipates system dynamics
  • Computed torque method compensates for nonlinear effects
  • Dynamic model compensation reduces tracking errors
• Feedback control integration corrects residual errors
  • PID control adjusts for unforeseen disturbances
  • Adaptive control techniques handle changing parameters
• Real-time considerations ensure practical implementation
  • Sampling rate affects control resolution
  • Computational efficiency enables faster response times

Stability analysis of PID systems

• Stability analysis techniques ensure system robustness
  • Routh-Hurwitz criterion analyzes characteristic equation
  • Root locus method visualizes stability regions
  • Nyquist stability criterion evaluates frequency response
• Performance metrics quantify system behavior
  • Steady-state error measures long-term accuracy
  • Percent overshoot indicates maximum deviation
  • Settling time quantifies stabilization period
  • Bandwidth determines system responsiveness
• Robustness analysis assesses stability margins
  • Gain margin indicates allowable gain increase
  • Phase margin measures resistance to delay
• Disturbance rejection evaluates external influence handling
  • Sensitivity function measures error suppression
  • Complementary sensitivity function assesses noise attenuation
• Noise sensitivity considers high-frequency effects
  • Derivative term can amplify measurement noise
• Limit cycles and oscillations may occur in nonlinear systems
  • Causes include actuator saturation and sensor nonlinearities
  • Prevention strategies involve gain adjustment and anti-windup techniques