Ballbot: A Single-Wheeled Balancing Robot | Dynamic Mobility

Control theory addresses making systems behave as desired. A system is any device or phenomenon composed of interconnected parts, from simple heaters to complex robots. Open-loop control lacks feedback, operating blindly without knowing actual output, creating problems illustrated by manual heaters and time-based traffic lights. Closed-loop control introduces feedback by measuring output and comparing to reference, enabling adaptive control actions. Sensors are essential but introduce limitations: measurable quantities must exist, and sensor accuracy determines performance. The two primary goals are stability (reliable predictable behavior) and tracking accuracy (output matching desired reference). Transfer functions describe input-output relationships in the Laplace domain, enabling mathematical analysis. For interconnected systems, overall transfer functions are products of individual components. The closed-loop transfer function Y = F/(1+F) × R determines how references propagate to outputs. System stability requires all poles (denominator roots) to have negative real parts. Controller design requires matching degrees of freedom to achieve desired pole placement. Controllability determines if states can be influenced by inputs, while observability determines if they affect outputs. Hidden states (neither controllable nor observable) cannot be changed by the controller. Any linear system can be transformed into Kalman canonical form, separating states into controllable, observable, and hidden blocks. The Routh-Hurwitz criterion provides a necessary condition for stability: all coefficients of the characteristic polynomial must have the same sign.

Control loops are essential systems that use feedback to maintain desired output parameters by continuously measuring actual values, comparing them to setpoints, and adjusting inputs accordingly; key performance metrics include load regulation (maintaining output despite load changes), line regulation (maintaining output despite input variations), and transient response (recovering quickly from disturbances without overshoot or oscillation); system stability is critical, as oscillations that persist beyond one or two cycles indicate an unstable system that cannot function reliably.

This section covers core control system concepts: (1) Transfer function is the ratio of output to input in Laplace domain with zero initial conditions, expressed as C(s)/R(s); (2) Self loop is a feedback path where start and end points are identical; (3) Steady state error is the error at t→∞, calculated as e_ss = lim(s→0) [sR(s)/(1+G(s)H(s))]; (4) System stability depends on pole locations in the S-plane: stable when all poles are in left-half plane, unstable when any pole is in right-half plane, marginally stable when poles are on imaginary axis.

System stability is determined by eigenvalue real parts: positive → unstable, negative → stable, zero → marginal/indeterminate. Transfer functions relate to state space through poles (denominator roots) and zeros (numerator roots); poles determine stability. Controllability (ability to reach any state via inputs) requires full-rank controllability matrix [B AB A²B...]. Observability (ability to infer states from outputs) requires full-rank observability matrix [C CA CA²...]. These concepts form the foundation of control theory alongside stability.

Control systems maintain outputs at desired states through feedback mechanisms. Key concepts include: control system purpose (maintaining outputs at target values), feedback control (measuring output and feeding back as input), open vs closed loop control, PID control components (proportional, integral, derivative), disturbances (unwanted external influences), steady-state error (residual difference after stabilization), response speed (time to reach targets), actuators (converting signals to physical actions), feedback loop components (sensors, controllers, actuators), negative feedback (reducing errors), overshoot (output exceeding target), gain (input amplification), stability (maintaining bounded operation), state space representation (state equations X_dot = AX + BU, output equations Y = CX + DU), state variables (mathematical representation of internal conditions), controllability (ability to drive system to any state), observability (ability to estimate states from outputs), and stability analysis methods (eigenvalues, Lyapunov theory). Advanced control design methods include: state feedback control (using all state variables), pole placement (designing desired dynamic response), optimal control (minimizing cost functions), LQ control (quadratic cost minimization for linear systems), Kalman filters (estimating states from noisy measurements), H-infinity control (providing robustness against disturbances and model uncertainties), adaptive control (automatically adjusting parameters for changing conditions), and model predictive control (predicting future behavior and optimizing inputs while handling constraints).

Torque is a force that gives rise to rotational movement. The torque is equal to the force F multiplied by the distance r from the fixed point (τ = F × r). Real massive bodies behave as if all their mass were concentrated at one point called the center of mass. Where a body is symmetrical in shape and either uniform in density or has a density that changes in symmetrical fashion, the center of mass is at the geometrical center. When a shell containing an explosive is moving at constant velocity and explodes, the explosion is an internal force and should have no effect on the motion of the system. The center of mass of the system moves on at constant velocity regardless of the explosion. When a body is falling toward the Earth, every particle is being pulled by gravity, but the body behaves as if all that force were concentrated at one point called the center of gravity. In a uniform gravitational field, the center of gravity would be identical with the center of mass. However, the lower portion of a body is closer to the center of the Earth than the upper portion, so the center of gravity is very slightly below the center of mass.

The center of mass is a particularly convenient reference point for rigid body dynamics. The acceleration of the center of mass depends only on external forces, and the rate of change of angular momentum about the center of mass equals the external torque about the center of mass. For uniform gravitational fields, the weight force acts at the center of mass, producing zero torque about this point. This simplifies analysis of rigid body dynamics under gravity.

This lesson covers rigid body mechanics: (1) Rigid bodies maintain shape under forces and undergo translational (straight-line) or rotational (circular) motion; (2) Torque causes rotation and equals force times lever arm; (3) Moment of inertia measures rotational resistance; (4) Angular momentum (L = Iω) is conserved without external torques; (5) Equilibrium requires zero net force and zero net torque; (6) Center of gravity is where resultant gravitational forces act; (7) Three equilibrium types: stable (returns to original position), unstable (moves further away), and neutral (remains in new position). Applications include bicycle balance, wrench torque, figure skater rotation, and spinning tops.

The fundamental equations of motion for rigid bodies are: sum of forces = dP/dt (linear momentum) and sum of torques = dL/dt (angular momentum). Different sets of forces can produce the same motion if they have the same net force and net torque. A resultant force is a single force that produces the same net force and net torque as a system of multiple forces. The weight of an extended body can be treated as a single force acting at the center of mass, where R_cm = (∫r dm) / M. In uniform gravitational fields, the center of gravity coincides with the center of mass. In non-uniform fields, the center of gravity is defined by R_cg = (∫r × dF) / (∫dF), which differs from the center of mass.

In rigid body mechanics, General Plane Motion (GPM) consists of two fundamental types: translation (caused by force imbalance) and rotation (caused by moment imbalance). Translation occurs when the net force on a rigid body is not zero, causing linear motion. Rotation occurs when the net moment about a point is not zero, causing angular motion. The center of gravity (CG) is the mean location of gravitational force acting on a body, while the center of mass (CM) is the mean location of all mass in a body. In a uniform gravitational field, CG and CM coincide. The moment of inertia represents a body's resistance to angular acceleration. Moments (torque) are calculated as the cross product of position vector and force vector, with direction determined by the right-hand rule. Objects tip over when the vertical line through their center of gravity falls outside their base of support.

Inertial Measurement Units (IMUs) are sensors that measure orientation and angular velocity using accelerometers and gyroscopes. The MPU6050 is a six-axis IMU containing both accelerometer and gyroscope sensors for three axes each. In robotics, IMUs enable robots to maintain stability by measuring their actual angle in space rather than relying solely on joint position feedback. This allows robots to compensate for unexpected tilts or disturbances by adjusting motor positions based on real-time spatial orientation data.

An Inertial Measurement Unit (IMU) combines accelerometer, gyroscope, and magnetometer data through sensor fusion algorithms to determine device orientation (pitch, yaw, roll) and provide absolute acceleration values in North-East-Down coordinates; the implementation requires calibrating the magnetometer by taking median values during rotation, using an AHRS algorithm like the Mahony Quaternion Update to fuse sensor data into quaternions, converting quaternions to rotation matrices, transforming acceleration vectors to absolute coordinates, and applying magnetic declination correction for accurate heading.

An IMU (Inertial Measurement Unit) combines multiple sensors to detect changes in orientation and position. The gyroscope measures rotational speed across three axes, useful for tracking precise turns. The accelerometer detects orientation relative to Earth's gravity, useful for detecting if the robot is tilted. The magnetometer (compass) detects Earth's magnetic field but is often disabled because motors create interfering magnetic fields. For most robotics applications, the gyroscope and accelerometer are most important. IMUs can be calibrated using provided SDK examples, though good results are often achievable out of the box.

An Inertial Measurement Unit (IMU) is a sensor module that combines multiple sensors to measure a robot's motion and orientation; it includes a 3-axis gyroscope for rotation detection, a 3-axis accelerometer for acceleration measurement, and a 3-axis magnetometer for magnetic field sensing, with common applications including determining heading (z-axis rotation), roll (y-axis rotation), pitch (x-axis rotation), and gravitational acceleration (typically 9.8 m/s² when stationary).

IMU sensor fusion combines accelerometer and gyroscope data to overcome individual sensor limitations. Accelerometers provide stable absolute orientation references but suffer from motion-induced errors during movement. Gyroscopes provide accurate short-term orientation changes but drift over time due to numerical integration. A complementary filter combines these sensors using low-pass filtering for accelerometer data and high-pass filtering for gyroscope data, eliminating both drift and short-term errors. IMUs alone cannot determine heading (yaw) around the vertical axis because gravity provides no reference for this rotation. AHRS (Attitude and Heading Reference Systems) solve this by adding three-axis magnetometers that measure Earth's magnetic field for drift-free heading reference. Combined with accelerometers and gyroscopes, AHRS provides complete 3D orientation measurement. These systems integrate signal processing to fuse sensor data, and can be combined with GPS and barometers for enhanced navigation.