Self-Balancing Robot Mechanics: Inverted Pendulum Control Theory

Sensor fusion is the process of combining data from multiple sensors to improve system performance, where the optimal combination of independent sensor measurements with different variances follows the formula z = ax₁ + bx₂ with coefficients a = σ₂²/(σ₁²+σ₂²) and b = σ₁²/(σ₁²+σ₂²), achieving lower variance than either individual sensor; this principle extends to Kalman filtering, which combines sensor measurements with system models using a recursive algorithm that calculates an optimal gain coefficient to optimally weigh model predictions against actual measurements, enabling improved accuracy in applications like navigation and tracking systems.

Sensor fusion combines information from multiple sensors measuring different physical quantities using system models, sensor equations, and dynamics equations. The Kalman filter handles measurements in different units by comparing estimated values from physical models with actual sensor measurements. For static systems, recursive moving averages converge toward true values following the law of large numbers. For dynamic systems, simple averaging fails because measurements from different times correspond to different states. The Kalman filter operates through alternating prediction and update steps: prediction applies system dynamics and commands to predict the next state with increasing uncertainty, while update incorporates measurements to reduce uncertainty using a Kalman gain that balances predicted estimates against measurements based on their relative reliability.

Kalman filtering is a mathematical technique that combines data from multiple sensors to produce more accurate state estimates by recursively predicting the system state using a dynamic model and then updating this prediction with sensor measurements, where each measurement is weighted according to its uncertainty (represented by covariance matrices), allowing devices like smartphones and drones to accurately determine their position and orientation by integrating information from complementary sensors such as GPS, accelerometers, and gyroscopes.

The Kalman Filter is an optimal estimation algorithm developed by Professor Dr. Rudolf Kalman for estimating system parameters from measurements. It consists of two stages: prediction and update. Stanley Smith applied it during Apollo missions to estimate engine temperatures (5500°F) using external sensors since internal sensors couldn't withstand extreme heat. Three main sensors are used for position tracking: GPS (1 Hz update rate, works outdoors only), IMU with gyroscope, accelerometer, and magnetometer (high frequency but accumulates drift errors), and odometry (depends on wheel-ground contact). Each sensor has different uncertainties. Sensor fusion combines multiple sensors to reduce overall uncertainty. Three main types of Kalman Filters exist: Extended Kalman Filter (EKF) for nonlinear problems using linearization, Unscented Kalman Filter (UKF) for nonlinear problems using sigma point transformations, and standard Kalman Filter for linear problems.

Sensor fusion combines multiple data sources to create improved estimates. Key advantages include: (1) Improved data quality by reducing noise through independent sensor combinations; (2) Increased reliability through redundancy to handle sensor failures; (3) Ability to estimate unmeasurable states using multiple perspectives; (4) Extended sensing coverage. The Kalman filter provides a mathematical framework for optimal sensor fusion, handling asynchronous measurements arriving at different times and frequencies. It calculates optimal weights (gain) for combining measurements based on their uncertainties, providing statistically optimal estimates under linear models and Gaussian noise assumptions. The filter operates through prediction and update steps, with careful attention to sensor independence and avoiding simultaneous failure modes.

La estabilidad en sentido de Lyapunov es un marco teórico fundamental para analizar la estabilidad de equilibrios en sistemas dinámicos no lineales, donde se define que un equilibrio es estable si soluciones iniciales cercanas permanecen cerca del equilibrio, asintóticamente estable si además tienden al equilibrio, y exponencialmente estable si lo hacen con una tasa exponencial. El método indirecto de Lyapunov utiliza la linealización del sistema alrededor del equilibrio y el análisis de los autovalores de la matriz jacobiana para determinar estabilidad o inestabilidad, mientras que el método directo emplea funciones de Lyapunov (definidas positivas cuya derivada temporal es semidefinida negativa) para establecer condiciones suficientes de estabilidad sin resolver explícitamente las ecuaciones diferenciales.

The Lyapunov stability theorem provides mathematical conditions for determining the stability of equilibrium points in nonlinear dynamical systems using a scalar function called a Lyapunov function. For an equilibrium point x*, if there exists a positive definite function v(x) (with v(x*)=0 and v(x)>0 elsewhere) whose time derivative v_dot(x) is negative semi-definite (≤0), then x* is stable; if v_dot(x) is negative definite (<0), then x* is locally asymptotically stable with a region of attraction defined by the domain of v(x); and if the domain is the entire state space and v(x) approaches infinity as ||x|| increases, then x* is globally asymptotically stable.

Nonlinear control theory focuses on analyzing dynamical systems defined by state equations x_dot = f(x) and determining conditions under which the system's state converges to an equilibrium point; the primary analytical tool is the Lyapunov function method, which uses a positive definite function V(x) whose time derivative along system trajectories must be negative definite to guarantee asymptotic stability, providing a systematic approach to stability analysis and controller design without requiring explicit solution of the differential equations.

Consider the nonlinear system x_dot = [x₁ + x₂, -x₁ - x₂ - x₁³ - x₂³]. Using V = x₁² + x₂² as a candidate Lyapunov function, we compute V̇ = 2x₁(x₁ + x₂) + 2x₂(-x₁ - x₂ - x₁³ - x₂³). Simplifying yields V̇ = -2x₂² - 2x₁² - 2x₁⁴ - 2x₂⁴, which is negative definite. Thus, the origin is asymptotically stable. This demonstrates how Lyapunov's method can analyze nonlinear stability that would be difficult to determine through other means.

Lyapunov's stability theorems provide a method to analyze the stability of nonlinear systems without solving the differential equations directly. The first theorem states that if there exists a continuously differentiable scalar function V (Lyapunov function) such that V(0) = 0, V(x) > 0 for all x ≠ 0 in a neighborhood, and its derivative V_dot ≤ 0, then the equilibrium point is stable. If V_dot < 0 (negative definite), the system is asymptotically stable. For global asymptotic stability, the Lyapunov function must also be radially unbounded (V(x) → ∞ as ||x|| → ∞). The derivative V_dot is calculated using the chain rule: V_dot = (∂V/∂x) × f(x). A practical example demonstrates that the choice of Lyapunov function significantly affects the strength of stability conclusions, as different functions can yield different stability results for the same system.

Hardware setup requires the microcontroller, programmer device, and computer with programming software. AVR Programmer software allows reading/writing fuse bits and programming hex files. Fuse bits control critical parameters: clock source (internal/external), frequency, and reset behavior. For ATtiny13, fuse bits determine internal 8 MHz oscillator or external quartz crystal usage. Memory constraints vary significantly: ATtiny13 has limited memory for simple programs, while ATmega8/16 have more resources. Microcontrollers support multiple communication protocols: SPI for high-speed devices, I2C for sensors, RS232 for serial communication, CAN for automotive applications, and UART for serial interfaces. Practical applications include audio channel switchers using relays to connect multiple input sources to single outputs.

Effective analog integration requires understanding signal chain construction and monitoring strategies. With eight-channel interfaces, you can build chains connecting multiple analog units (compressors, EQs, saturators) in series. Each unit processes the signal sequentially before returning to the original input pair. Bypass switches allow instant comparison between processed and unprocessed signals. For interfaces with limited I/O, alternative monitoring methods using dedicated tracks enable hardware processing. These techniques maximize the utility of your analog gear within a digital workflow.

Sloppy design proves invaluable when integrating 3D printed parts with existing hardware that doesn't meet specifications. The presenter demonstrates this through an outdoor electronic door controller project, where a chicken door system couldn't lift the 750-gram door it was designed for. Solutions include: designing control boxes with sliding clamping systems for horizontal alignment, creating mounting holes with five millimeter clearance to accommodate misaligned holes, using washers to compensate for poor alignment, and designing motor mounts with slots instead of holes for horizontal adjustment. The presenter also shows how to create symmetrical slots using construction lines and mirror tools, and how to add recesses for bolt alignment. This approach allows for fine-tuning during installation, compensating for hardware limitations, measurement errors, and installation challenges.

Hardware integration advances include auto-rotate support for cameras like Logitech StreamCam, projector preview configuration flexibility, multi-source copy-paste functionality, Twitch extension emote support for BBTV and FFZ, and Intel Ice Lake Quick Sync encoder optimizations with custom quantization matrices. Practical workflow features include scroll filter looping options, expanded filename formatting variables for resolution and framerate, hotkey for stats panel reset to eliminate benchmarking artifacts, and increased default text/source sizes for better visibility.

The WaveShare 2.7-inch e-paper HAT supports both Arduino and Raspberry Pi platforms. For Arduino, connect 3.3V, ground, MOSI (D11), SCK (D13), CS (D10), DC (D9), RST (D8), and BUSY (D7). Use libraries like 'e-paper' or 'SoonUse' from GitHub, installing them in the Arduino libraries directory. For Raspberry Pi, enable SPI through Preferences > Raspberry Pi Configuration, then run Python examples from the cloned GitHub repository. Both platforms demonstrate similar functionality including text, shapes, and icons.