Mathematics for Robotics

ROB-GY 6013 A · NYU Tandon · Fall 2026

Wednesdays 5:00–7:30 PM · 2 MetroTech Center, Room 909 · Zoom for anyone remote

Staff

Jing ZhangInstructor z.jing@nyu.edu

Shichen ZhangTeaching assistant sz4968@nyu.eduOffice hours Fridays 12:30–1:30 PM, 6 MetroTech Center (in person) or Zoom (remote). See Ed post for updates.

Links

Course website Brightspace Ed Discussion Gradescope Notation Reading guide

Schedule

Date	Topic	Optional reading	Homework
Part I · Linear Algebra
Sep 2 Slides Interactive	M1Introduction and Linear Algebra I · Spaces, Maps, Rank, Solutions	Grizzle ROB 501 notes 04: subspaces and linear independence Grizzle ROB 501 notes 05: basis vectors and dimension Grizzle ROB 501 notes 03: abstract linear algebra Grizzle ROB 501 notes 01: introduction and proofs Chiang, Rush and Barak: Named Tensor Notation (TMLR 2023) 11 videos Strang 18.06 L10: the four fundamental subspaces Strang 18.06 L6: column space and nullspace 3Blue1Brown EoLA 2: span and basis vectors 3Blue1Brown EoLA 1: vectors 3Blue1Brown EoLA 3: linear transformations Grizzle ROB 501 L4: subspaces and linear independence Grizzle ROB 501 L5: basis vectors and dimension Strang 18.06 L8: solving Ax = b, the row reduced form R 3Blue1Brown EoLA 7: column space and null space Strang 18.06 L7: pivot variables and special solutions Strang 18.06 L9: independence, basis, dimension	Practice set 1 · not turned in
Sep 9 Slides Interactive	M2Linear Algebra II · Coordinates, Determinant, Eigenstructure	Grizzle ROB 501 notes 06: linear operators and eigenvalues Axler, Linear Algebra Done Right, 4th edition (free, open access) 14 videos Strang 18.06 L30: linear transformations and their matrices Strang 18.06 L31: change of basis 3Blue1Brown EoLA 13: change of basis Strang 18.06 L18: properties of determinants Strang 18.06 L20: Cramer's rule, the inverse matrix, and volume Modern Robotics 3.2.1: rotation matrices (part 1 of 2)3Blue1Brown EoLA 6: the determinant Strang 18.06 L21: eigenvalues and eigenvectors Strang 18.06 L22: diagonalization and powers of A 3Blue1Brown EoLA 14: eigenvectors and eigenvalues Strang 18.06 L28: similar matrices and Jordan form Strang 18.065 L8: norms of vectors and matrices Grizzle ROB 501 L6: linear operators and eigenvalues Strang 18.06 L24: Markov matrices; Fourier series	Practice set 2 · not turned in
Sep 16 Slides Interactive Recording	M3Linear Algebra III · Orthogonality, Projection, Least Squares, QR	Grizzle ROB 501 notes 17: Gram-Schmidt vs modified Gram-Schmidt Stanford EE263 notes: least-squares and the orthogonality principle Stanford EE263 notes: QR factorization by Gram-Schmidt 13 videos Harvard AM205 2.8: Gram-Schmidt, classical and modified Harvard AM205 2.9: Householder triangularization Harvard AM205 2.10: Givens rotations 3Blue1Brown EoLA 9: dot products and duality Strang 18.06 L14: orthogonal vectors and subspaces Strang 18.06 L17: orthogonal matrices and Gram-Schmidt Strang 18.065 L3: orthonormal columns Stachniss: least squares, an informal introduction Stachniss SLAM 15: least squares SLAM Strang 18.06 L15: projections onto subspaces Strang 18.06 L16: projection matrices and least squares Strang 18.065 L9: four ways to solve least squares Strang 18.065 L10: difficulties with Ax = b	Practice set 3 · not turned in
Sep 23 Slides Interactive Recording	M4Linear Algebra IV · PSD, SVD, PCA, Pseudoinverse, Jacobian and Hessian	Grizzle ROB 501 notes 11: symmetric and orthogonal matrices Grizzle ROB 501 handout: the SVD and numerical rank of a matrix Tedrake, Robotic Manipulation ch. 3: differential kinematics, the Jacobian pseudo-inverse, and singularities Stanford EE263 notes: the SVD, matrix gain, pseudo-inverse, and condition number Rycroft, AM205 Unit 2 notes: numerical linear algebra (conditioning, QR, SVD) 15 videos Harvard AM205 2.7: QR decomposition and why not the normal equations ALAFF 1.4.1: the condition number of a matrix Strang 18.06 L4: factorization into A = LU Strang 18.065 L2: multiplying and factoring matrices Stachniss: ICP and point cloud registration, part 1 (known correspondences, SVD)Stachniss: the numerics of bundle adjustment Strang 18.06 L25: symmetric matrices and positive definiteness Strang 18.06 L29: singular value decomposition Strang 18.065 L7: Eckart-Young, the closest rank-k matrix Strang 18.06 L33: left and right inverses, the pseudoinverse Strang 18.065 L5: positive definite and semidefinite matrices Strang 18.06 L27: positive definite matrices and minima Modern Robotics 5.3: singularities Modern Robotics 5.1.1: the space Jacobian Modern Robotics 6.2: numerical inverse kinematics	PS 1 · due Oct 7, 5 PM
Part II · ML Math
Sep 30 Slides Interactive	M5ML Math I · Matrix Calculus and Backprop	CS231n backprop Baydin et al.: Automatic Differentiation in Machine Learning, a Survey CS231n Neural Networks 3: gradient checks, centred differences and relative error Parr & Howard: The Matrix Calculus You Need For Deep Learning TorchCode: implement layers and their backward passes, with auto-grading (MIT licence) 3 videos 3Blue1Brown DL 1: what is a neural network 3Blue1Brown DL 3: backpropagation, intuitively 3Blue1Brown DL 4: backpropagation calculus
Oct 7 Slides Interactive LayerNorm backward Attention backward	M6ML Math II · Autograd, Optimization, and Neural Building Blocks	Grizzle ROB 501 notes: convex sets and convex functions EE364a slides Boyd & Vandenberghe: Convex Optimization, the full text free from the authors 1 video MIT 18.065 L25: stochastic gradient descent	PS 1 due 5 PMPS 2 · ps2-autograd.ipynbdue Oct 21, 5 PM
Oct 14	No class — Legislative Day, and the three prior offerings did not make it up
Part III · Dynamics and Probability
Oct 21 Slides Interactive Worked solutions (PDF)	M7ODE and Simulation · Continuous Dynamics and Numerical Integration	Tedrake, Underactuated Robotics Ch. 2: the simple pendulum, phase portraits and fixed points 4 videos Strang 18.06 L23: differential equations and exp(At)Harvard AM205 3.10: ODE stability Harvard AM205 3.11: Runge–Kutta methods Modern Robotics 8.1: Lagrangian formulation of dynamics (part 1 of 2)	PS 2 due 5 PM
Oct 28	📝 Midterm — the whole meeting, in person in Room 909, covering M1 through M6
Nov 4 Slides Interactive Worked solutions (PDF)	M8Probability I · Random Variables, Bayes, Gaussian, Estimation	Grizzle ROB 501 notes: Gaussian random vectors Grizzle ROB 501 notes: probability space 10 videos Stanford CS229 L5: the multivariate Gaussian in GDA Tuebingen Probabilistic ML L6: Gaussian conditioning and marginalisation MIT 6.041 L8: continuous random variables and the PDF Grizzle ROB 501: Gaussian random vectors Grizzle ROB 501: probability space and random variables CMU 10-708 L9: Gaussian graphical models and the precision matrix MIT 18.102 L7: sigma-algebras 3Blue1Brown: Bayes theorem, the geometry of changing beliefs Stachniss SLAM-Course 03: the Bayes filter MIT RES.6-012 L16.3: squared error asks for the conditional mean	PS 3 out · due Nov 18, 5 PM
Nov 11 Slides Interactive Worked solutions (PDF)	M9Probability II · Information Theory, Sampling, and Density	Blanchard, Higham & Higham: accurate computation of log-sum-exp and softmax MacKay, Information Theory, Inference, and Learning Algorithms (free book) 5 videos 3Blue1Brown: information theory and entropy (from 8:04)Kuleshov, Cornell CS6785 L4: maximum likelihood as KL Rigollet, MIT 18.650 (Fall 2016) L4: total variation and KL divergence Abbeel, Berkeley CS294-158 (Spring 2020) L10: compression, prefix codes, Kraft de Freitas, UBC CPSC 540 (2013): importance sampling and MCMC I
Part IV · Robot Policies
Nov 18 Slides Interactive Worked solutions (PDF)	M10Transformers and ACT · Implementation and Action Chunking	ACT, Zhao et al. 2023: the CVAE latent, the KL term, and action chunking The Annotated Transformer: the paper line by line in PyTorch Attention Is All You Need (Vaswani et al. 2017): the primary paper Shazeer 2019, Fast Transformer Decoding: One Write-Head Is All You Need — the KV cache as a memory-bandwidth cost 8 videos Stanford CS224N L8: self-attention and transformers Stanford CS336 Lec 3: architectures and RoPE LeRobot #1: ALOHA and ACT walkthrough CMU RI Seminar: action chunking (Simchowitz)Stanford MLSys #67: FlashAttention (Tri Dao)Stanford CS224R L2: imitation learning (Finn)Raschka L8.8: softmax derivatives for gradient descent Stanford CS336 Lec 10: inference, the KV cache and the cost of decoding	PS 3 due 5 PMPS 4 out · due Dec 2, 5 PM
Nov 25 Slides Interactive Worked solutions (PDF)	M11Diffusion and Diffusion Policy · Score-Based Models to Robot Policies	Diffusion Policy, Chi et al.: p(a_{t:t+H} \| o_t)Yang Song's blog: score-based generative modelling MIT 6.S184: diffusion and flow matching, slides and notes Stanford CS236 Deep Generative Models: course slides Abbeel, Berkeley CS294-158 Deep Unsupervised Learning (Spring 2024): course slides DDPM, Ho et al. 2020: the primary diffusion paper DDIM, Jiaming Song et al. 2020: non-Markovian sampling in few steps 14 videos Abbeel, Berkeley CS294-158 (Spring 2024) L6: diffusion models Stanford CME296 L1: DDPM training, DDPM inference, faster sampling with DDIM Stanford CME296 L2: score matching Yang Song at MIT CBMM: diffusion and score-based generative models Stanford CME296 L4: classifier-based guidance, then classifier-free guidance Cheng Chi: Diffusion Policy, talk by the paper's first author Russ Tedrake at Princeton: dexterous manipulation with diffusion policies Miika Aittala (EDM co-author): sampling design and the 35-evaluation budget MIT 18.S096 L21: stochastic differential equations, the OU process solved in closed form Stanford CS236 L12: energy-based models MIT RES.6-012 L16.3: squared error asks for the conditional mean (callback to M8)Wen Sun, Cornell CS4789 L17: behaviour cloning and distribution shift Kuleshov, Cornell CS6785 L6: REINFORCE vs reparameterization MIT 6.S184 (2025) L5: diffusion for robotics — guest lecture, Benjamin Burchfiel (TRI)
Dec 2 Slides Interactive Worked solutions (PDF)	M12Flow Matching and VLA · Flow Matching to π₀ and π₀.₅	Flow Matching Guide and Code: the long-form tutorial π₀, Physical Intelligence: a VLM plus a flow-matching action expert Flow Matching for Generative Modeling (Lipman et al. 2022): the primary paper Rectified Flow — Flow Straight and Fast (Liu et al. 2022): straight paths and the coupling of π₀ and π₁ 8 videos MIT 6.S184 L02: flow matching MIT 6.S184 L01: flows, ODEs, and the flow map MIT 6.S184 L03A: score functions, and converting score to velocity Brunton: the continuity equation from the divergence theorem Brunton: Euler error via Taylor series Levine at U of T: pi-zero's action expert and the 50-step chunk Adams: optimal transport as a linear program Peyre: entropic regularization and Sinkhorn (10:54-24:05)	PS 4 due 5 PM
Dec 9	📝 Final examination — the whole meeting, in person in Room 909. The date is pending departmental confirmation; hold December 16–22 as well until it is confirmed.