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Mathematics for Autonomous Systems

This section is intended for robotics engineers and software developers requiring a rigorous framework for autonomous behavior. The following modules cover the core disciplines of Dynamical Systems, Linear Algebra, and Lie Theory, alongside modern approaches in Optimization and Machine Learning. This curriculum establishes the formal language necessary to model physical constraints and manage environmental uncertainty. Mastery of these subjects is a prerequisite for designing and implementing high-performance algorithms for perception, estimation, and real-time motion planning.

%% TITLE: FIGURE 1: Simplified Robotics Domain Map
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%% 1. The Math Basics
subgraph Foundations
    DS[Dynamical Systems]:::subject
    LS[Linear Systems]:::subject
    PROB[Probability]:::subject
    NUM[Numerical Methods]:::subject
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%% 2. The Advanced Toolset
subgraph Advanced Topics
    OPT[Optimization]:::subject
    LIE[Lie Theory]:::subject
    ML[Machine Learning]:::subject
end

%% 3. The Robotic Capabilities
subgraph Applications
    EST(Estimation):::hub
    PLAN(Planning):::hub
    PERC(Perception):::hub
    DEC(Decision Making):::hub
end

Outline

Dynamical Systems

Linear and nonlinear ordinary differential equations; Initial Value Problems and numerical integration; Phase Portraits and Equilibrium points; Laplace and Z-transforms; Transfer Functions; effects of poles and zeros on frequency, and stability analysis.

Systems theory provides the mathematical infrastructure for modeling and analyzing systems.

  • Inverted Pendulum Simulation: Derive the equations of motion using Lagrangian mechanics and simulate the nonlinear system. Linearize around the upright/downright equilibrium points to analyze their stability.
  • Robot Chassis Simulation: Develop a mobile robot chassis model. Start with known constraints (size, weight,...) and then design the parameters of the motors and wheels.
  • Frequency Response Analysis: Tune a generic 2nd order transfer function to fit the bode plot of an "unknown" system.

Linear Systems

Vector spaces and norms; matrices and linear transformations; rank, nullity, and the four fundamental subspaces; eigenvalues and eigenvectors; diagonalization and Jordan normal form; orthogonality and projections; least-squares fitting; determinants and inverses; QR, SVD, and LU decompositions; condition numbers and numerical stability.

Nearly every algorithm in robotics reduces to a linear algebra operation. You cannot design a Kalman filter, invert kinematics, optimize trajectories, or compress data without it.

  • Camera Calibration: Recover intrinsic and extrinsic parameters from checkerboard images via least-squares regression.
  • Dynamic Mode Decomposition: Recover the discrete linear dynamics of a system by exciting it and measuring the states.
  • Hardware Optimized BLAS: Implement a custom matrix multiplication function that utilizes a SIMD or NEON hardware feature. Use this function to simulate and analyze a system. Examine the precision and performance between a naive gemm function and a parallelized one.

Probability and Information Theory

Probability axioms and rules; conditional probability and Bayes' rule; random variables and distributions (Gaussian, exponential, Laplace); expectation, variance, and covariance; independence and conditional independence; entropy and mutual information; relative entropy (KL divergence); Fisher information and the Cramér-Rao bound; parameter estimation (maximum likelihood, MAP); Bayes filters and recursive estimation; Kalman filters and Extended Kalman Filters; particle filters and Monte Carlo localization.

You cannot fuse sensor data, estimate hidden states, or make principled decisions under uncertainty without probability theory. It is essential for perception, localization, and planning in the real world.

  • Monte Carlo localization: Implement a particle filter for a mobile robot in a known map with laser rangefinder and odometry.
  • Kalman Filter for Inertial Measurement: Fuse accelerometer and gyroscope data to estimate 3D orientation with explicit covariance propagation.
  • GPS/Odometry Fusion for Localization: Implement an Extended Kalman Filter (EKF) to fuse noisy GPS measurements with wheel odometry data for robust 2D pose estimation of a mobile robot, including covariance management and outlier rejection.

Numerical Methods

Root-finding (bisection, Newton-Raphson, secant method); linear system solvers (Gaussian elimination, iterative methods); eigenvalue computation (power iteration, QR algorithm); numerical integration of ODEs (Euler, RK2, RK4, symplectic integrators); error analysis and convergence rates; stability of difference equations; stiffness and implicit methods; constraint handling (penalty methods, Lagrange multipliers); numerical optimization (line search, trust regions, Hessian approximation).

This topic relates advanced linear algebra tools and computer architecture.

  • Satellite Orbit Simulation: Integrate rigid-body dynamics (nonlinear ODEs) forward in time to predict motion, using RK4 or symplectic integrators for energy stability.
  • Manipulator Pose Planning: Iteratively solve a nonlinear system (Newton-Raphson) to find joint angles that achieve a desired end-effector pose.

Lie Theory

Lie groups and Lie algebras; SO(3) and SE(3); exponential and logarithmic maps; tangent spaces and differential; geodesics and geodesic distance; quaternions and their relationship to SO(3); adjoint representations; perturbation analysis; manifold optimization; numerical integration on manifolds.

Ad-hoc use of Euler angles will lead to singularities and numerical instability. Proper use of Lie groups ensures your rotation computations are singularity-free, differentiable, and efficient.

  • Spherical Linear Interpolation (SLERP): Interpolate between two 3D rotations via the geodesic in SO(3), preserving angular velocity.
  • Kinematic Control on SO(3): Implement an attitude feedback controller (3-loop autopilot, PID, LQR) to stabilize a quadcopter's attitude.
  • Manifold Sampling for Motion Planning: Generate collision-free configurations sampling directly on SE(3), avoiding singularities.

Optimization

Unconstrained optimization (gradient descent, Newton's method, quasi-Newton methods, line search, trust regions); convex optimization (convex sets and functions, duality, KKT conditions); constrained optimization (Lagrange multipliers, penalty and augmented Lagrangian methods); linear programming; quadratic programming; semidefinite programming; sequential convex programming; gradient-free methods (genetic algorithms, simulated annealing); convergence analysis and rates.

You'll spend most of your time formulating problems and choosing solvers. Understanding convexity, duality, and KKT conditions helps you recognize solvable problems and detect non-convexity before wasting compute on bad local minima.

  • Trajectory Optimization: Design a transfer orbit for a satellite to change its orbit altitude. Assume an ideal elliptical orbit at both altitudes, the input is the current orbit, target orbit and the output is the transfer path (plus entry/exit points).
  • Physics Informed Neural Network: Use a neural network to solve partial differential equations by embedding the physics equations as a regularization term in the loss function.

Machine Learning

Linear and polynomial regression; regularization (L1, L2); classification (logistic regression, SVMs, decision trees); neural networks and backpropagation; convolutional and recurrent architectures; unsupervised learning (k-means, GMM, PCA); dimensionality reduction (t-SNE, auto-encoders); clustering; Bayesian inference and expectation-maximization; reinforcement learning (Markov decision processes, value iteration, policy gradient); model evaluation and validation.

Modern perception (object detection, pose estimation, semantic segmentation) relies entirely on learned models. Data-driven control and planning are increasingly competitive with classical methods.

  • Object Detection and 6D Pose: Fine-tune a CNN for detecting objects and regressing their 3D pose from RGB-D images for robotic grasping.
  • Dynamics Model Learning: Learn a forward model (state + action → next state) from robot trajectories to enable black box model-based planning and control.
  • Reinforcement Learning for Navigation: Train a policy via deep Q-learning or policy gradients to navigate a mobile robot to a goal while avoiding obstacles.