Mathematical Foundations for Robotics Systems Engineering: Linear Algebra, Multivariable Calculus, and Lie Groups for Motion, Estimation, and Dynamics

Build robotics models you can trust by mastering the mathematics that governs motion, estimation, and rigid-body geometry.

This engineering-focused text develops the essential tools behind modern robotic systems, starting from core linear algebra and multivariable calculus and extending through dynamics, discretization, and uncertainty. The emphasis is always on sensitivity, stability, and validation, so results are not only derived correctly but also checked for conditioning, error growth, and physical consistency.

Inside you will work through the mathematical backbone of robotics, including:

• Linear algebra for real-world solves: rank and null spaces, QR and Cholesky structure, eigendecomposition, SVD, pseudoinverses, regularization, and condition numbers
• Multivariable calculus that drives kinematics and estimation: Jacobians, Hessians, chain rules, and implicit differentiation for constrained mechanisms
• Nonlinear least squares and constrained optimization: Gauss-Newton, Levenberg-Marquardt, Lagrange multipliers, KKT conditions, and quadratic programs
• Differential equation modeling and discretization: state-space ODEs, matrix exponential solutions for LTI systems, sampling effects, truncation error, and stiffness-aware integration ideas
• Uncertainty and error propagation: first-order sensitivity, covariance propagation through nonlinear maps, and Mahalanobis geometry
• Rigid-body geometry with Lie groups: SO(3) and SE(3), exponential and logarithm maps, left and right Jacobians, adjoints, and consistent perturbation models
• Kinematics and dynamics fundamentals: product of exponentials forward kinematics, Jacobians and singularities, spatial vector dynamics, and Lagrangian structure
• Estimation foundations: linear Kalman filtering and manifold-aware extensions using tangent-space error models

Every chapter includes multiple choice questions plus practice questions with full answers, designed to reinforce ＂derive, solve, validate＂ skills. The result is a rigorous, problem-driven pathway to the mathematical foundations needed for reliable robotics systems engineering.