Control of Mobile Robots | Differential Drive Kinematics & Unicycle Model

Non-holonomic constraints are constraints that cannot be expressed as simple scalar equations relating only the generalized coordinates; instead, they involve relationships between the generalized velocities (time derivatives of coordinates). The general form is Σ(a_li(q) × q̇_i) = 0 for l = 1 to S, where S is the number of non-holonomic constraints. Unlike holonomic constraints, these constraints restrict instantaneous velocities rather than configurations. A classic example is rolling without slipping, where the point of contact between a wheel and ground must have zero velocity relative to the surface—a constraint on velocity rather than position.

Holonomic constraints are expressible as equations g_i(r₁, r₂, ..., r_N, t) = 0, reducing DOF by integers. Examples: rigid rods (|r₁ - r₂| - l = 0), particles on tables (y = 0), rolling without slipping (s = rθ). Non-holonomic constraints involve inequalities or differential conditions and don't reduce DOF by integers. Rolling without slipping couples translational and rotational motion through v = ω×r. The general formula for DOF is f = dN - r, where d is dimension and r is constraint count. This mathematical framework enables systematic analysis of constrained mechanical systems.

This section establishes the complete mathematical framework for analyzing mechanical constraints. It covers: (1) Classification of constraints as holonomic (integrable to coordinate-time equations) or non-holonomic (velocity-dependent); (2) The integrating factor criterion for determining if a non-holonomic constraint can be reduced to holonomic form; (3) The mathematical conditions requiring mixed partial derivatives of μf to be equal; (4) Application to the rolling disk constraint, showing how ẋ - Rφ̇ = 0 can be integrated to x - Rφ = constant. The section demonstrates both integrable and irreducible cases, providing the theoretical foundation for analyzing constraint systems in analytical mechanics.

This extensive section develops the complete mathematical framework for analyzing constrained mechanical systems, spanning from theoretical foundations to practical applications. It begins by expressing constraints in linear non-holonomic (Pfaffian) form: Σ(a_li(q) * dq_i/dt) + a_lt(q) = 0, where virtual displacements δq_i replace differentials with dt = 0 for instantaneous analysis. Lagrange multipliers λ_l are introduced for each constraint, transforming the problem into a system where δq_i can vary independently while satisfying constraints. The derivation combines D'Alembert's principle with constraint summation to yield modified Lagrange's equations: d/dt(∂L/∂q̇_i) - ∂L/∂q_i = Q_i + Σ_l=1^s (λ_l * a_li(q)). The system now has n Lagrange equations plus s constraint equations, totaling n+s equations for n+s unknowns. The generalized forces of constraint relate to Newtonian forces through projection vectors: C_i = Σ_j [F_j^constraint · γ_ji + M_j^constraint · β_ji]. The section then explores non-holonomic systems through the roller racer example, demonstrating how two knife edge constraints define hyperplanes in 4-dimensional velocity space, whose intersection forms a 2-dimensional accessible subspace. Despite having only one control input, the system achieves complex motions including translation and rotation, illustrating the counterintuitive property that fewer controls can achieve more complex maneuvers in non-holonomic systems.

This tutorial demonstrates how to improve robot odometry accuracy by fusing wheel encoder odometry with laser scan matcher odometry using an Extended Kalman Filter (EKF). Wheel encoders suffer from drift due to wheel slip, while laser scan matchers struggle in featureless environments. The EKF automatically adjusts weights between sensors based on their reliability, giving more weight to wheel encoders in corridors and more weight to laser odometry during rotation. The tutorial covers implementing RF2O laser odometry, configuring the EKF filter, and visualizing the fused odometry in RViz2 to demonstrate the improvement in robot position estimation.

The Extended Kalman Filter can be applied to robot localization, where the robot tracks its position using odometry and landmark observations. The EKF works well for moderate nonlinearities and bounded uncertainties but fails with highly nonlinear models or multimodal priors. Alternative approaches like the Unscented Kalman Filter offer different approximation strategies. The Kalman filter is the optimal recursive Bayesian estimator for Gaussian linear systems, implementing prediction-correction steps using mean and covariance updates. The EKF extends this to nonlinear systems through local linearization via Taylor expansions, replacing constant matrices with Jacobians evaluated at the predicted mean.

Autoware implements a layered localization architecture: Earth frame receives GNSS data, Global State Estimator produces earth-frame poses, Transform lookup converts to map frame, Environmental Sensor Localizer (NDT) matches lidar against pre-built maps, Map State Estimator combines estimates for map-to-odom transform, Semantic Map hosts HD maps separately, and Odometry State Estimator processes IMU/wheel encoders for odom-to-base_link. Base Link sits at rear axle center for sensor offset measurement, while Nav Base sits at center of gravity for control purposes. Kalman filters (EKF/UKF) implement state estimation through predict-correct cycles, with robot_localization handling multi-sensor fusion (10+ sensors with varying update rates and qualities). Sensor covariance and process covariance configuration is critical for accurate estimation, with wheel encoders providing good velocity but poor orientation, and IMUs providing good orientation but poor position.

The Extended Kalman Filter (EKF) is a recursive algorithm used for estimating the state of a nonlinear system, particularly useful in robot localization where the robot's position and orientation must be estimated based on noisy sensor measurements and motion models; the EKF extends the standard Kalman Filter by linearizing the nonlinear system dynamics and observation models around the current estimate using Taylor series expansion, allowing it to handle the nonlinearity inherent in robot motion and sensor measurements.

Sensor fusion improves robot localization accuracy by combining multiple sensor data streams (such as wheel odometry and IMU) using algorithms like Kalman filters, which helps compensate for individual sensor limitations such as wheel slippage on low-friction surfaces; this approach provides more reliable and accurate robot state estimation compared to relying on a single sensor type.

A differential drive robot model maps electrical inputs (voltages) to wheel angular velocities through motor dynamics, then to robot pose (position and orientation) through kinematic relationships, enabling prediction of robot motion and determination of required wheel commands for desired movement; however, all models are simplifications that make assumptions (such as pure rolling without slipping) to achieve practical utility.

The ideal differential drive model uses wheel radius and baseline (distance between contact points) as parameters. For skid steering vehicles, extended models account for different instantaneous centers of rotation for each wheel's contact point with the ground. The asymmetric model uses separate left and right wheel slip parameters (alpha_L, alpha_R). The symmetric model assumes symmetrical centers of rotation relative to the robot body frame, adding an estimated baseline parameter and a single wheel slip parameter. The full linear model adds beta_1 and beta_2 parameters to capture the relationship between center distances and the robot body origin.

A dynamic model describes the relationship between a robot's motions and the forces that cause the motion. Unlike kinematic models, dynamic models include forces and torques. The general form is τ = M(q)q̈ + C(q, q̇)q̇ + G(q) + F(q, q̇), where τ is applied torque, M is the inertia matrix, q̈ is acceleration, C represents centripetal/coriolis forces, G is gravitational torque, and F represents frictional forces.

The dynamic model for the Mecanum wheel robot includes propulsion forces that can act in both X and Y directions, as well as torques. The force and torque vector (talco) captures translational and rotational dynamics. Coriolis terms appear (negative for Y, positive for X) and damping terms can be included in both directions depending on their significance. For rotation, friction and damping terms (beta_R and beta_RR) represent resistance forces opposing angular motion. The torque must overcome these resistance forces to achieve angular acceleration.

To simulate real-world conditions, wheel slipping can be modeled by adding a gain block (1-s) where s is the slipping percentage (e.g., 2%). This gain multiplies the velocity commands before integration to calculate position. Different slipping percentages can be applied to each wheel to simulate asymmetric traction conditions.

This video explains how to use binary occupancy grids for motion planning in autonomous navigation. Motion planning involves finding a trajectory connecting a robot's starting state to a goal state, with path planning focusing on the sequence of poses rather than motion derivatives. Two main approaches exist: search-based algorithms like A* (which prioritizes nodes using a heuristic combining actual cost and straight-line distance to goal) and sampling-based algorithms like RRT and RRT* (which build trees by randomly sampling the state space and connecting nodes to nearest neighbors, with RRT* additionally rewiring connections to approach optimal paths). These graph-based methods discretize continuous environments into nodes and edges, enabling efficient path finding even in complex, high-dimensional state spaces.

RRT* improves upon RRT by rewiring connections to nearby nodes within a radius, achieving asymptotic optimality as samples increase. Bidirectional RRT variants grow trees from both start and goal configurations toward each other. Modern motion planning combines multiple approaches: search-based methods for low-dimensional problems, sampling-based methods for high-dimensional spaces, learning-based approaches for adaptation, and optimization-based techniques for refinement. Hybrid strategies leveraging complementary strengths represent the state-of-the-art in robotics motion planning, particularly for complex manipulators requiring both feasibility and optimality.

Path planning finds optimal vehicle trajectories under constraints using map data and vehicle position. The Navigation Toolbox provides self-localization, SLAM, and path planning algorithms including A* and RRT. Hybrid A* extends A* for vehicles with kinematic constraints by considering vehicle dynamics. Implementation involves loading map data, creating vehicle state-space models (x, y, θ), generating occupancy grids, creating obstacle objects with validation distances, configuring the planner with start/goal positions, and generating paths. Results can be visualized and implemented in Simulink for code generation.

Motion planning algorithms help autonomous robots find obstacle-free paths from start to goal locations; common approaches include grid-based methods like A* and hybrid A* for holonomic robots, and sampling-based methods like PRM and RRT for non-holonomic robots with differential constraints, where RRT explores state space incrementally by randomly sampling nodes and expanding toward them, while RRT* improves upon RRT by rewiring the tree to find optimal shortest paths.

Rapidly Exploring Random Trees (RRT) are a class of probabilistic motion planning algorithms that efficiently explore high-dimensional spaces by growing a tree from a start node toward randomly sampled points; RRT* improves upon basic RRT by rewiring the tree to find shorter paths, achieving asymptotic optimality at the cost of increased computation, while all variants are probabilistically complete meaning they will find a path if one exists.