The Robotics Checklist

1.1 Linear Algebra

• Vectors and vector spaces — the basic object; positions, velocities, forces, and torques are all vectorsexplainer ↗
• Dot product — projection and alignment; work is force dotted with displacementexplainer ↗
• Cross product — perpendicular vector, area, and the source of torque and angular velocity relationships
• Matrix multiplication — composition of transformations; the operation every kinematics chain is made of
• Matrix transpose and inverse — undoing a transformation, and why orthogonal matrices are cheap to invert
• Determinant — volume scaling, and zero determinant as the signal of a singularity
• Rank and null space — how many independent directions a mechanism can actually move in
• Eigenvalues and eigenvectors — principal axes of inertia, stability of linear systems, vibration modesexplainer ↗
• Singular Value Decomposition (SVD) — manipulability analysis, least squares, pseudo-inverse for redundant arms
• Moore–Penrose pseudo-inverse — solving underdetermined and overdetermined systems; core to redundant IK
• Damped least squares / Levenberg–Marquardt — pseudo-inverse that doesn't explode near singularities
• Positive definite matrices — inertia matrices, covariance matrices, and what makes an energy function valid
• Quadratic forms — kinetic energy, cost functions, ellipsoids of manipulability and uncertainty
• Skew-symmetric matrices — the matrix form of a cross product; central to rotation derivatives
• Homogeneous coordinates — adding a dimension so that translation becomes a matrix multiplication
• Numerical conditioning — why a nearly-singular Jacobian produces enormous joint velocities

1.2 Rotations and Rigid Transformations

• Why orientation is hard — rotations don't commute, don't add, and can't be represented globally by three numbers without a singularity
• Rotation matrices and SO(3) — orthonormal columns, determinant +1, the canonical representation
• Euler angles — roll-pitch-yaw and the dozen other conventions; intuitive, compact, and singular
• Gimbal lock — losing a degree of freedom when two axes align; the reason Euler angles fail in flight control
• Axis-angle representation — a unit axis and a rotation about it; compact and geometric
• Rodrigues' rotation formula — converting axis-angle to a rotation matrix in closed form
• Quaternions — four numbers, no singularities, efficient composition; the practical default for orientation
• Quaternion double cover — q and −q represent the same rotation, and the bugs that causes
• SLERP — spherical linear interpolation; the correct way to blend between two orientations
• Homogeneous transformation matrices and SE(3) — rotation and translation in one 4×4 object
• Transform composition and inversion — chaining frames, and getting the order right
• Frame conventions — body vs world, active vs passive, pre- vs post-multiplication; the source of endless sign errors
• Lie groups and Lie algebras — SO(3), SE(3) and their tangent spaces so(3), se(3)
• Exponential and logarithm maps — moving between a velocity in the algebra and a pose in the group
• Twists — linear and angular velocity as a single six-vector
• Wrenches — force and torque as a single six-vector; the dual of a twist
• Screw theory and the product of exponentials — a coordinate-free alternative to DH parameters
• Adjoint transformations — mapping twists and wrenches between frames

1.3 Calculus and Differential Geometry

• Derivatives and the chain rule — the backbone of every gradient and every Jacobian
• Partial derivatives and gradients — how a cost changes with each joint
• Jacobian matrices — the linear map from joint velocities to end-effector velocity; the single most important derivative in robotics
• Hessians — curvature, used in second-order trajectory optimization
• Taylor expansion and linearization — how nonlinear robot dynamics get turned into a linear controller
• Ordinary differential equations — the language every dynamic system is written in
• Numerical integration — Euler, RK4, semi-implicit and symplectic integrators; why simulators drift
• Stability of ODEs — equilibrium points, phase portraits, basins of attraction
• Manifolds and tangent spaces — why you cannot naively average two rotations
• Geodesics — shortest paths on a curved space; the right notion of "interpolate between orientations"
• Variational calculus — the mathematics behind Lagrangian mechanics and optimal control

1.4 Probability and Estimation

• Random variables and distributions — sensor noise is not a nuisance, it's a modelling object
• Gaussian distributions — the workhorse; multivariate form, covariance ellipsoids
• Conditional probability and Bayes' theorem — the entire basis of state estimation
• Marginalization — integrating out what you don't care about
• Covariance matrices — encoding uncertainty and its correlations across state dimensions
• Maximum likelihood and MAP estimation — the two standard ways to fit a state or a parameter
• Least squares and weighted least squares — the deterministic face of Gaussian estimation
• Markov assumption — the next state depends only on the current one; makes filtering tractable
• Hidden Markov Models — discrete-state estimation from noisy observations
• Monte Carlo methods — sampling instead of solving; the basis of particle filters
• Information matrix and information filter — the inverse-covariance view, natural for sparse SLAM
• Chi-squared test and Mahalanobis distance — outlier rejection and data association gating
• RANSAC — fitting a model when a large fraction of your data is wrong

1.5 Optimization

• Convexity — why some problems solve reliably and others don't
• Gradient descent — the fundamental iterative method
• Newton and quasi-Newton methods — using curvature; Gauss-Newton and Levenberg–Marquardt in particular
• Constrained optimization and Lagrange multipliers — joint limits, obstacle constraints, torque bounds
• KKT conditions — the general optimality conditions with inequality constraints
• Linear programming — scheduling, allocation, some contact problems
• Quadratic programming (QP) — the workhorse of real-time control; whole-body control solves a QP every cycle
• Sequential Quadratic Programming (SQP) — nonlinear trajectory optimization
• Nonlinear programming solvers — IPOPT, SNOPT, OSQP, qpOASES and what each is good for
• Mixed-integer programming — discrete decisions like footstep placement or contact mode selection
• Complementarity problems (LCP) — the mathematical form contact and friction take
• Sampling-based optimization — CEM, MPPI; gradient-free and increasingly common in MPC
• Real-time optimization constraints — warm starting, anytime algorithms, and hard deadlines

1.6 Numerical Methods and Computation

• Floating point representation — precision limits, and why accumulating rotations drifts
• Numerical stability and conditioning — when small input errors become large output errors
• Sparse matrices — SLAM and multibody problems are enormously sparse and exploiting that is essential
• Matrix factorizations — LU, QR, Cholesky; which to use for which structure
• Iterative solvers — conjugate gradient and friends for very large systems
• Automatic differentiation — exact derivatives without hand-deriving Jacobians; CasADi, JAX, autodiff in Drake
• Fixed-point arithmetic — still relevant on microcontrollers without an FPU
• Computational complexity in real-time contexts — an O(n³) algorithm in a 1 kHz loop is a design error

2.1 Classical Mechanics

• Newton's laws — the foundation; F = ma and its rotational counterpart
• Reference frames — inertial vs non-inertial, and the fictitious forces that appear in rotating frames
• Kinematics of a point — position, velocity, acceleration and their relationships
• Work, energy and power — the currency of actuator sizing
• Conservation laws — energy, momentum, angular momentum
• Impulse and momentum — the right tools for impacts and collisions
• Centre of mass — and why it matters for balance, tipping, and manipulator payload
• Moments and torques — force applied at a distance; the quantity motors actually produce
• Statics and free-body diagrams — the analysis skill everything mechanical rests on
• Friction — static vs kinetic, Coulomb model, stiction, and why it's the hardest thing to simulate

2.2 Rigid Body Dynamics

• Rigid body assumption — what it buys and when it breaks
• Inertia tensor — the rotational analogue of mass; a 3×3 matrix, not a scalar
• Principal axes and moments of inertia — the eigen-decomposition of the inertia tensor
• Parallel axis theorem — shifting an inertia tensor to a different reference point
• Angular velocity and angular acceleration — and why angular velocity is not the derivative of Euler angles
• Euler's equations of motion — rotational dynamics of a rigid body
• Newton–Euler formulation — force and torque balance applied link by link
• Recursive Newton–Euler Algorithm (RNEA) — O(n) inverse dynamics; the standard fast method
• Articulated Body Algorithm (ABA) — O(n) forward dynamics
• Composite Rigid Body Algorithm — efficient computation of the mass matrix
• Spatial vector algebra — Featherstone's 6D notation that makes these algorithms clean

2.3 Analytical Mechanics

• Generalized coordinates — describing configuration with the minimum number of variables
• Degrees of freedom — counting them correctly, including Grübler/Kutzbach criteria
• Constraints — holonomic vs nonholonomic, and why a car is harder to plan for than a drone
• Lagrangian mechanics — kinetic minus potential energy; derive equations of motion without free-body diagrams
• Euler–Lagrange equations — the resulting differential equations
• The manipulator equation — M(q)q̈ + C(q,q̇)q̇ + g(q) = τ; the single most important equation in manipulator control
• Mass/inertia matrix M(q) — configuration-dependent, symmetric, positive definite
• Coriolis and centrifugal terms C(q,q̇) — velocity-dependent coupling between joints
• Gravity vector g(q) — the term gravity compensation cancels
• Hamiltonian mechanics — the energy-based formulation, used in passivity-based control
• Passivity and energy shaping — control designed so the closed loop cannot generate energy
• Virtual work and d'Alembert's principle — the bridge between statics and dynamics

2.4 Contact, Friction and Impact

• Contact modelling — rigid vs compliant contact, and the tradeoffs of each in simulation
• Coulomb friction cone — the constraint that tangential force is bounded by normal force
• Stick-slip transitions — the discontinuity that makes contact simulation numerically nasty
• Restitution and impact models — what happens in the instant of collision
• Complementarity formulation of contact — either there's a gap or there's a force, never both
• Penalty methods vs constraint methods — spring-damper contact vs exact non-penetration
• Contact-rich manipulation — insertion, sliding, pivoting; where models are least trustworthy
• Grasp force closure and form closure — the conditions under which a grasp actually holds

2.5 Structures and Materials

• Stress and strain — the basic quantities of structural analysis
• Young's modulus and material stiffness — the property that sets deflection
• Beam bending and deflection — cantilever arms sag, and by how much matters
• Yield strength and factor of safety — designing so things don't break
• Fatigue and cyclic loading — robots do the same motion a million times; static analysis isn't enough
• Buckling — slender members failing in compression
• Natural frequency and resonance — structural modes that limit achievable control bandwidth
• Damping — and why an under-damped structure makes a control engineer's life miserable
• Material selection — aluminium, steel, carbon fibre, plastics; strength-to-weight and cost
• Finite element analysis (FEA) — numerical stress and modal analysis

2.6 Fluids and Aerodynamics

• Fluid statics and buoyancy — underwater vehicles
• Drag and lift — the forces on any body moving through a fluid
• Reynolds number — the regime indicator that tells you which physics dominates
• Propeller and rotor aerodynamics — thrust and torque as functions of RPM; the basis of multirotor control
• Ground effect and vortex ring state — the aerodynamic failure modes drones actually hit
• Pneumatics — compressible flow, valves, cylinders; ubiquitous in industrial automation
• Hydraulics — incompressible power transmission; high force density, used in heavy machinery and some legged robots

2.7 Energy and Thermal

• Power budgeting — the calculation that determines whether your robot runs for 20 minutes or 8 hours
• Battery chemistry and characteristics — Li-ion, LiPo, LiFePO4; energy density, discharge curves, C-rating
• Battery management systems (BMS) — cell balancing, protection, state of charge estimation
• Efficiency chains — every conversion loses energy; motor, gearbox, driver, and battery all take a cut
• Heat generation and dissipation — motors and drivers get hot, and thermal limits often bind before torque limits
• Thermal derating — continuous vs peak torque ratings and what actually determines them
• Regenerative braking — recovering energy, and where it goes if the battery won't take it

3.1 Circuit Fundamentals

• Voltage, current, resistance — Ohm's law and what each quantity physically is
• Kirchhoff's laws — current and voltage conservation; the basis of all circuit analysis
• Power dissipation — I²R losses and why wire gauge matters
• Capacitors and inductors — energy storage, time constants, and transient behaviour
• RC, RL and RLC circuits — filtering and the frequency response of real wiring
• Diodes — rectification, flyback protection, and why an inductive load needs one
• Transistors: BJT and MOSFET — switching and amplification
• Operational amplifiers — signal conditioning, buffering, instrumentation amplifiers
• Voltage regulators — linear vs switching, and the efficiency tradeoff
• Grounding and star grounding — the single most common cause of mysterious noise
• Decoupling capacitors — and why leaving them out produces intermittent, unreproducible faults

3.2 Power Electronics

• H-bridge — the circuit that lets a DC motor run in both directions
• PWM (pulse width modulation) — controlling average power with a switching duty cycle
• Three-phase inverters — driving BLDC and PMSM motors
• Gate drivers — turning MOSFETs on and off fast enough to be efficient
• Dead time — the deliberate delay that stops both halves of a bridge conducting at once
• Current sensing — shunt resistors, Hall-effect sensors; needed for torque control
• Buck, boost and buck-boost converters — DC-DC conversion topologies
• Inrush current and soft start — why closing a contactor onto a capacitive load can weld it
• Bus voltage and regeneration — where the energy goes when a motor decelerates
• Braking resistors and clamp circuits — dumping regenerated energy safely
• Fusing and circuit protection — coordination, interrupt ratings, and protecting the wire not the load

3.3 Motors and Actuators

• Brushed DC motors — simple, cheap, and the brushes wear out
• Brushless DC (BLDC) and PMSM — higher efficiency and power density; require electronic commutation
• Field-oriented control (FOC) — the standard method for smooth, efficient BLDC torque control
• Commutation and the Clarke/Park transforms — the coordinate changes that make FOC work
• Stepper motors — open-loop positioning, and the risk of silently losing steps
• AC induction motors — the industrial workhorse; VFD control
• Servo motors and servo drives — closed-loop position/velocity/torque control as an integrated unit
• Torque constant and back-EMF constant — the two numbers that define a motor's electrical behaviour
• Motor speed-torque curves — continuous vs peak operating regions
• Motor sizing — inertia matching, duty cycle, thermal RMS torque calculation
• Series elastic actuators (SEA) — a deliberate spring in the drivetrain for force control and safety
• Quasi-direct drive actuators — low gear ratio for backdrivability; the modern legged robot approach
• Harmonic drive / strain wave gearing — high ratio, zero backlash, expensive
• Cycloidal drives — high ratio, high stiffness, high shock tolerance
• Planetary gearboxes — compact, coaxial, the common general-purpose choice
• Backlash — lost motion at direction reversal; the enemy of precision and of stable control
• Backdrivability — whether an external force can move the joint; central to safe physical interaction
• Pneumatic actuators — cylinders, valves, air preparation; fast, compliant, hard to control precisely
• Hydraulic actuators — enormous force density, high maintenance, used in heavy machinery
• Shape memory alloys and soft actuators — emerging alternatives for soft robotics

3.4 Sensors and Signal Conditioning

• Encoders: incremental — quadrature counting, index pulse, and losing position on power cycle
• Encoders: absolute — single-turn and multi-turn; position known at boot
• Resolvers — rugged analogue position sensing for harsh environments
• Hall effect sensors — commutation feedback and simple position sensing
• Potentiometers — cheap absolute position, limited life and resolution
• IMUs: accelerometers and gyroscopes — and the bias, drift, and noise characteristics of each
• Magnetometers — heading reference, and how easily ferrous structures corrupt it
• Force/torque sensors — six-axis wrist sensors; strain gauge based, expensive, drift with temperature
• Current-based torque estimation — the cheap alternative to a force sensor, with friction as the error term
• Tactile sensors — resistive, capacitive, optical (GelSight-style); still an open problem
• Proximity sensors — inductive, capacitive, photoelectric; the bread and butter of industrial sensing
• Limit switches and homing — establishing an absolute reference at startup
• Ultrasonic and infrared range sensors — cheap distance, poor angular resolution
• Analogue-to-digital conversion — resolution, sample rate, aliasing
• Nyquist sampling theorem — sample at more than twice the highest frequency or see false signals
• Anti-aliasing filters — a hardware requirement, not an optional extra
• Sensor noise characterization — Allan variance for IMUs; knowing your noise is prerequisite to filtering it
• Calibration — scale, bias, misalignment, temperature compensation
• Signal filtering — low-pass, complementary, and the latency cost every filter imposes

3.5 Embedded Electronics and EMC

• Microcontroller peripherals — GPIO, timers, ADC, DAC, PWM, DMA, interrupts
• Level shifting and isolation — optocouplers, digital isolators, and protecting logic from power
• PCB design basics — trace width, layer stackup, return paths, connector selection
• Electromagnetic interference (EMI/EMC) — motors are enormous noise sources sitting next to your sensors
• Shielding and cable routing — separating power and signal, twisted pairs, shield termination
• Common-mode vs differential signalling — why RS-485 and CAN use differential pairs
• Ground loops — and the isolation strategies that break them
• ESD protection — and the failures that appear weeks after the actual discharge

4.1 Mechanisms

• Kinematic pairs and joints — revolute, prismatic, spherical, cylindrical, planar, screw
• Serial vs parallel mechanisms — an arm vs a Stewart platform or delta robot
• Four-bar linkages — the fundamental mechanism; Grashof condition
• Cam mechanisms — converting rotation into arbitrary motion profiles
• Differential mechanisms — combining or splitting motion between outputs
• Compliant mechanisms — flexure-based motion with no sliding joints
• Overconstraint and kinematic determinacy — designing so assembly doesn't fight itself
• Common arm configurations — articulated, SCARA, cartesian/gantry, delta, cylindrical, spherical
• Wrist configurations — spherical wrists and why they simplify inverse kinematics

4.2 Transmission and Drivetrain

• Gear ratios — trading speed for torque, and the effect on reflected inertia
• Reflected inertia — load inertia divided by the square of the gear ratio; drives motor selection
• Belt and pulley drives — timing belts, tensioning, compliance
• Chain drives — high load, needs lubrication
• Ball screws and lead screws — rotary to linear; efficiency and back-drivability differ enormously
• Linear guides and rails — the precision constraint on any linear axis
• Cable and tendon drives — remote actuation, used in surgical robots and hands
• Bearings — ball, roller, thrust, plain; preload, life calculation
• Couplings — rigid, flexible, and accommodating misalignment
• Slip rings — passing power and signal across a continuously rotating joint

4.3 Design and Manufacturing

• CAD proficiency — SolidWorks, Fusion 360, Onshape, FreeCAD; parametric modelling and assemblies
• Tolerance analysis — stack-up, and why a chain of "close enough" parts doesn't assemble
• GD&T (geometric dimensioning and tolerancing) — communicating what actually matters to a machinist
• Design for manufacture (DFM) — designing parts that can actually be made economically
• Design for assembly (DFA) — and for disassembly, because robots need servicing
• Machining processes — milling, turning, and what each can and can't produce
• 3D printing — FDM, SLA, SLS; prototype vs production suitability, anisotropic strength
• Sheet metal design — bend radii, relief cuts, and cost-effective enclosures
• Fasteners and joining — thread standards, preload, thread locking, and why things vibrate loose
• Ingress protection (IP) ratings — sealing against dust and water
• Cable management — drag chains, service loops, and the flex life of a cable that bends a million times
• Weight budgeting — especially binding for mobile and aerial platforms
• CAE and simulation — FEA for structure, modal analysis for vibration, MBD for mechanism motion

5.1 Forward Kinematics

• Kinematic chains — links and joints, and the tree or chain structure they form
• Frame assignment — attaching a coordinate frame to every link
• Denavit–Hartenberg (DH) parameters — the classical four-parameter convention
• Standard vs modified DH — two incompatible conventions in wide use; check which a source means
• Product of exponentials (PoE) — screw-theory forward kinematics; no frame assignment needed
• URDF and robot description formats — how a robot model is actually specified in software
• Tool centre point (TCP) and tool frames — where the robot thinks the useful point is
• Base and world frames — and the calibration that relates them to reality
• Forward kinematics for mobile bases — odometry from wheel motion

5.2 Inverse Kinematics

• The IK problem — given a desired pose, find joint angles; generally harder than forward kinematics
• Existence and multiplicity of solutions — zero, several, or infinitely many
• Analytical/closed-form IK — exact, fast, and only available for specific geometries
• Pieper's criterion — three consecutive axes intersecting makes closed-form IK possible
• Numerical IK — Jacobian transpose, pseudo-inverse, damped least squares
• Cyclic Coordinate Descent (CCD) — simple iterative IK, common in animation
• FABRIK — a fast heuristic IK method
• Optimization-based IK — posing IK as a constrained nonlinear program; handles joint limits and secondary objectives
• IK solvers in practice — IKFast, TRAC-IK, KDL, BioIK and their tradeoffs
• Joint limits and self-collision — the constraints that make a mathematically valid solution physically useless
• Solution branch selection — elbow up vs elbow down, and the continuity problems of switching mid-trajectory

5.3 Velocity Kinematics

• The geometric Jacobian — mapping joint velocities to end-effector twist
• The analytical Jacobian — the same idea using a specific orientation parameterization
• Jacobian computation — column by column from screw axes, or by differentiating forward kinematics
• Inverse velocity kinematics — solving for joint velocities given a desired end-effector velocity
• Static force relationship — τ = Jᵀ·F; the Jacobian transpose maps end-effector forces to joint torques
• Manipulability ellipsoid — visualizing how easily the arm can move in each direction
• Manipulability measure — a scalar quality metric derived from the Jacobian's singular values
• Singularities — configurations where the Jacobian loses rank and a direction of motion is lost
• Types of singularity — boundary, internal, and wrist singularities
• Singularity avoidance and damped inverses — keeping joint velocities bounded near a singularity
• Kinematic redundancy — more joints than task dimensions
• Null space projection — using redundancy for secondary objectives like avoiding joint limits or obstacles

5.4 Mobile Robot Kinematics

• Differential drive — two independently driven wheels; the simplest mobile base
• Nonholonomic constraints — a car cannot move sideways, and that changes planning fundamentally
• Ackermann steering — car-like kinematics and the bicycle model
• Omnidirectional drives — mecanum and omni wheels; holonomic motion at the cost of efficiency and traction
• Tracked vehicles — skid steering, and the slip that makes odometry unreliable
• Instantaneous centre of rotation (ICR) — the geometric construction underlying wheeled motion
• Wheel odometry — dead reckoning from encoders, and how quickly it drifts
• Slip and its effects — why odometry alone is never enough

6.1 Modelling

• Robot dynamics model — the manipulator equation and its terms
• Inverse dynamics — given a motion, what torques are needed
• Forward dynamics — given torques, what motion results; needed for simulation
• Dynamic parameter identification — estimating link masses, inertias and friction from measured data
• Friction models — Coulomb, viscous, Stribeck; identification and compensation
• Actuator dynamics — motor and gearbox behaviour that the rigid-body model ignores
• Joint flexibility — the dominant unmodelled effect in geared arms
• Model uncertainty — and designing controllers that tolerate it

6.2 Classical Control

• Feedback control concept — measure, compare, correct
• Open loop vs closed loop — and when open loop is genuinely the right answer
• Transfer functions and the Laplace domain — the classical analysis language
• Block diagrams and loop algebra — composing systems
• Poles and zeros — and what they mean for response speed and stability
• Stability criteria — Routh–Hurwitz, Nyquist, gain and phase margins
• Step response characteristics — rise time, overshoot, settling time, steady-state error
• P, PI, PD and PID control — what each term does and when to use it
• PID tuning — Ziegler–Nichols, relay auto-tuning, and manual tuning by feel
• Integral windup and anti-windup — the practical failure every PID implementation must handle
• Derivative kick and filtering — why you differentiate the measurement, not the error
• Feedforward control — using a model to act before the error appears
• Cascade control — nested position/velocity/current loops; the standard servo architecture
• Bode plots and loop shaping — frequency-domain design
• Bandwidth and its physical limits — structural resonance and sample rate cap what's achievable
• Discretization — z-transform, sample rate selection, and the delay a digital loop adds

6.3 State Space and Modern Control

• State space representation — ẋ = Ax + Bu, y = Cx + Du
• Controllability and observability — whether you can steer the state, and whether you can see it
• Pole placement — designing feedback gains to put closed-loop poles where you want them
• Linear Quadratic Regulator (LQR) — optimal state feedback for a quadratic cost
• Kalman filter as an observer — the dual of LQR
• LQG control — LQR plus a Kalman filter, and the robustness caveats
• Integral action in state space — eliminating steady-state error
• Observers and state estimation for control — Luenberger observers
• Robust control and H-infinity — designing for a bounded set of possible plants
• Multivariable control — coupling between axes and why SISO tuning fails on it

6.4 Nonlinear and Advanced Control

• Why robot dynamics are nonlinear — configuration-dependent inertia and velocity coupling
• Lyapunov stability theory — proving stability without solving the differential equations
• Feedback linearization — cancelling nonlinearity with an inverse model
• Computed torque control — the robotics-specific form of feedback linearization
• Gravity compensation — the simplest and most useful model-based term
• Sliding mode control — robust to model error, at the cost of chattering
• Backstepping — recursive Lyapunov-based design for cascaded systems
• Adaptive control — estimating uncertain parameters online
• Passivity-based control — guaranteeing the closed loop cannot inject energy
• Control barrier functions (CBFs) — enforcing safety constraints as a filter on any controller
• Gain scheduling — interpolating between linear controllers across the operating envelope

6.5 Optimal Control and MPC

• Optimal control formulation — minimize a cost over a trajectory subject to dynamics
• Pontryagin's maximum principle — the classical necessary conditions
• Dynamic programming and the HJB equation — the value-function view
• Model Predictive Control (MPC) — optimize over a receding horizon, apply the first action, repeat
• Linear MPC — a QP solved every cycle; widely deployed and well understood
• Nonlinear MPC — more capable, much harder to run in real time
• Differential Dynamic Programming (DDP) and iLQR — efficient trajectory optimization
• MPPI and sampling-based MPC — gradient-free, GPU-friendly, increasingly popular
• Constraint handling — the main practical reason to choose MPC over LQR
• Horizon length and computational budget — the central tuning tradeoff
• Warm starting and real-time iteration — making NMPC fit inside a control cycle

6.6 Interaction Control

• Why position control fails on contact — a stiff position controller against a rigid environment produces enormous forces
• Impedance control — regulating the dynamic relationship between motion and force
• Admittance control — the dual formulation; better suited to stiff, non-backdrivable robots
• Stiffness, damping and inertia shaping — the three parameters of a virtual mechanical impedance
• Hybrid force/position control — controlling force in constrained directions and position in free ones
• Direct force control — closing a loop on measured force
• Compliance: passive vs active — a physical spring vs a software one, and why passive is safer
• Contact stability and the passivity condition — why coupling a stiff controller to a stiff environment goes unstable
• Whole-body control — solving for all joint torques at once as a hierarchical QP

6.7 Trajectory Generation

• Point-to-point vs continuous path — two fundamentally different motion requirements
• Trapezoidal velocity profiles — the standard industrial motion profile
• S-curve profiles and jerk limiting — smoothing acceleration to reduce vibration and wear
• Polynomial trajectories — cubic and quintic splines through waypoints
• B-splines and NURBS — smooth parametric paths with local control
• Time-optimal trajectory generation — the fastest motion subject to actuator limits
• Time parameterization along a path (TOPP) — separating the geometric path from the timing
• Blending and cornering — passing near waypoints without stopping
• Joint space vs Cartesian space trajectories — and the different problems each causes
• Online trajectory generation — reacting within a control cycle, e.g. Reflexxes-style methods

7.1 Sensing Modalities

• Choosing a sensor suite — the design decision that constrains everything downstream
• Monocular cameras — cheap, dense, passive, and scale-ambiguous
• Stereo cameras — depth from disparity; fails on textureless surfaces
• RGB-D cameras — structured light and time-of-flight; excellent indoors, poor in sunlight
• 2D LiDAR — a planar scan; the classic indoor mobile robot sensor
• 3D LiDAR — spinning and solid-state; accurate range, sparse vertically, expensive
• Radar — works in rain, fog and dust; low resolution, direct velocity measurement via Doppler
• Ultrasonic — cheap, short range, wide beam; still used for bumper-level sensing
• Event cameras — per-pixel brightness changes at microsecond latency; excellent for high-speed motion
• Thermal cameras — sees heat, works in darkness
• Sensor comparison matrix — range, resolution, frame rate, cost, power, weather tolerance, failure modes

7.2 Camera Geometry and Calibration

• Pinhole camera model — the projection from 3D to 2D
• Intrinsic parameters — focal length, principal point, skew; the camera matrix K
• Lens distortion — radial and tangential; the correction that must happen before any geometry
• Extrinsic parameters — where the camera sits relative to the robot
• Camera calibration — checkerboard and ChArUco targets, Zhang's method
• Stereo calibration and rectification — aligning two cameras so disparity search is one-dimensional
• Hand-eye calibration — solving AX = XB to find the transform between a camera and a robot flange
• Epipolar geometry — essential and fundamental matrices, the epipolar constraint
• Triangulation — recovering a 3D point from two or more views
• PnP (Perspective-n-Point) — recovering camera pose from known 3D-2D correspondences
• Reprojection error — the standard objective for every geometric vision optimization

7.3 Classical Computer Vision

• Image representation and colour spaces — RGB, HSV, grayscale, and when each helps
• Convolution and filtering — blurring, sharpening, denoising
• Edge detection — Sobel, Canny
• Corner and blob detection — Harris, FAST, DoG
• Feature descriptors — SIFT, SURF, ORB, BRIEF; what makes a descriptor robust
• Feature matching — brute force, FLANN, ratio test, cross-checking
• RANSAC for geometric fitting — robust estimation with heavy outlier contamination
• Homography estimation — planar scene relationships; useful for ground-plane work
• Optical flow — Lucas–Kanade sparse and Farnebäck dense
• Morphological operations — erosion, dilation, opening, closing for binary cleanup
• Thresholding and segmentation — Otsu, adaptive thresholding, watershed
• Template matching — still the right answer for many controlled industrial inspection tasks
• Fiducial markers — AprilTag, ArUco, ChArUco; reliable pose from a printed pattern
• Blob analysis and connected components — the backbone of classical machine vision

7.4 Learned Perception

• Image classification — CNNs and vision transformers
• Object detection — YOLO, Faster R-CNN, DETR; boxes plus labels in real time
• Semantic and instance segmentation — per-pixel labels; Mask R-CNN, SAM
• Keypoint and pose estimation — human pose, and object 6D pose estimation
• 6D pose estimation for grasping — the perception output manipulation actually needs
• Depth estimation from monocular images — learned, scale-ambiguous, improving rapidly
• Open-vocabulary detection — CLIP-based and grounded detection; find objects you never trained on
• Visual foundation models — DINOv2, SAM and similar as general-purpose feature extractors
• Real-time constraints — model selection under a fixed latency budget; quantization, TensorRT, ONNX
• Edge deployment — Jetson, Coral, and the accuracy-latency-power tradeoff
• Domain shift in the field — the model was trained in a lab and the factory has different lighting
• Failure detection — knowing when perception is wrong is more valuable than being right more often

7.5 3D Perception

• Point clouds — the fundamental 3D data structure
• Point cloud filtering — voxel downsampling, statistical outlier removal, passthrough
• Normal estimation — surface orientation at each point
• Point cloud registration — ICP, Generalized ICP, NDT; aligning two scans
• Global registration — feature-based initial alignment before ICP refinement
• Plane and primitive fitting — RANSAC for planes, cylinders, spheres
• Segmentation and clustering — Euclidean clustering, region growing
• Occupancy grids and OctoMap — probabilistic 3D volumetric mapping
• Signed distance fields (SDF/TSDF) — implicit surface representation; the basis of KinectFusion-style mapping
• Meshes and surface reconstruction — Poisson reconstruction, marching cubes
• NeRF and Gaussian splatting — learned scene representations, increasingly used in robotics
• PCL and Open3D — the two main libraries and their tradeoffs

7.6 Sensor Fusion

• Why fuse — every sensor has a failure mode that another covers
• Complementary filters — the simple, cheap fusion method for IMU attitude
• Kalman-based fusion — the principled approach when you can model the noise
• Time synchronization — hardware triggering, PTP, and why unsynchronized sensors ruin fusion
• Extrinsic calibration between sensors — camera-to-LiDAR, camera-to-IMU
• Loosely vs tightly coupled fusion — fusing processed estimates vs raw measurements
• Data association — deciding which measurement corresponds to which object or landmark
• Degradation and fallback — behaving sensibly when a sensor drops out

7.7 Tactile and Force Perception

• Force/torque sensing at the wrist — the standard industrial approach
• Joint torque sensing — per-joint sensors, as in collaborative arms
• Tactile skins — distributed contact sensing over a surface
• Vision-based tactile sensors — GelSight and relatives; high spatial resolution from a camera behind a membrane
• Slip detection — knowing the object is escaping the grasp before it lands on the floor
• Contact state estimation — inferring what kind of contact is happening from force signatures

8.1 Bayesian Filtering

• The state estimation problem — recovering what the robot cannot directly measure
• The Bayes filter — predict with a motion model, correct with a measurement model, repeat
• Motion models — odometry model, velocity model, and their noise characteristics
• Measurement models — beam models, likelihood fields, landmark models
• Prediction and update steps — the two halves of every filter
• Belief representation — the choice that distinguishes every filter variant

8.2 The Kalman Family

• Kalman filter — the optimal linear-Gaussian estimator
• Process and measurement noise (Q and R) — the two matrices you will spend the most time tuning
• Innovation and Kalman gain — how much to trust the measurement versus the prediction
• Extended Kalman Filter (EKF) — linearizing a nonlinear system; the workhorse despite its flaws
• Unscented Kalman Filter (UKF) — sigma points instead of linearization; better on strong nonlinearity
• Error-state / indirect Kalman filter — the standard formulation for orientation, avoiding quaternion constraint issues
• Information filter — the inverse-covariance dual; sparse and natural for multi-sensor fusion
• Filter divergence — the failure mode where the filter becomes confidently wrong
• Consistency checking — NEES and NIS tests to verify your filter believes reasonable things

8.3 Nonparametric Filtering

• Particle filters — representing belief with weighted samples; handles multimodal and nonlinear cases
• Importance sampling and resampling — the core mechanics
• Particle deprivation — the failure where all particles collapse onto one wrong hypothesis
• Adaptive particle counts (KLD sampling) — spend particles where uncertainty demands it
• Monte Carlo Localization (MCL/AMCL) — the standard mobile-robot localization method
• Global localization and the kidnapped robot problem — recovering from complete loss of position
• Histogram and grid filters — discretized belief, simple and interpretable

8.4 SLAM

• The SLAM problem — build a map while localizing within it, with neither known in advance
• Why it's hard — the chicken-and-egg coupling and accumulating drift
• Full vs online SLAM — estimating the whole trajectory or just the current pose
• EKF-SLAM — historically important, scales quadratically with landmark count
• FastSLAM — particle filter over trajectories with per-particle landmark filters
• GraphSLAM and pose graph optimization — the modern standard; nodes are poses, edges are constraints
• Front end vs back end — data association and feature extraction versus optimization
• Loop closure detection — recognizing a previously visited place; the single most important correction
• Bag of visual words and place recognition — DBoW, NetVLAD
• Relocalization — recovering pose within an existing map
• Map representations — occupancy grids, feature maps, topological maps, TSDF, meshes
• LiDAR SLAM — Cartographer, LOAM, LIO-SAM, KISS-ICP
• Visual SLAM — ORB-SLAM3, RTAB-Map, and direct methods like DSO
• Visual-inertial odometry (VIO) — VINS-Fusion, OpenVINS; the standard for drones and AR
• Multi-session and lifelong mapping — maps that survive the environment changing
• Map maintenance — handling moved furniture, seasonal change, and construction

8.5 Optimization-Based Estimation

• Factor graphs — the modern unifying formulation for estimation problems
• Bundle adjustment — jointly optimizing camera poses and 3D structure
• Nonlinear least squares — Gauss-Newton and Levenberg-Marquardt on a sparse problem
• Sparsity structure — exploiting the fact that most poses don't see most landmarks
• Marginalization and sliding windows — bounding computation by dropping old states correctly
• Robust cost functions — Huber, Cauchy, Geman-McClure to survive bad data associations
• Libraries — GTSAM, g2o, Ceres Solver, and what each is suited to
• iSAM2 and incremental smoothing — updating a solution without re-solving from scratch

9.1 Foundations

• Configuration space (C-space) — the space of all robot configurations; where planning actually happens
• C-space obstacles — how a workspace obstacle becomes a much more complicated C-space region
• Free space and connectivity — what "a path exists" actually means
• Completeness — resolution complete, probabilistically complete, and neither
• Optimality and asymptotic optimality — will it find the best path, eventually or ever
• Collision checking — the operation that dominates planning runtime; broad phase and narrow phase
• Distance queries and swept volumes — checking motion between configurations, not just at endpoints
• FCL, Bullet and collision libraries — the standard implementations

9.2 Path Planning

• Graph search: BFS, DFS, Dijkstra — the foundations
• A\* — heuristic search; admissibility and consistency
• Weighted and anytime A\* — trading optimality for speed under a deadline
• D\* and D\* Lite — efficient replanning when the map changes
• Field D\* and Theta\* — any-angle paths that don't follow grid edges
• Grid and lattice representations — discretizing the world, and the resolution tradeoff
• State lattice planners — precomputed motion primitives respecting vehicle kinematics
• Hybrid A\* — continuous-state search for car-like vehicles; used in parking and autonomous driving
• Visibility graphs and Voronoi diagrams — classical geometric approaches
• Potential fields — elegant, fast, and prone to local minima
• Probabilistic Roadmaps (PRM) — build a graph by sampling; good for repeated queries in a static world
• Rapidly-exploring Random Trees (RRT) — grow a tree toward random samples; the sampling-based default
• RRT\* — asymptotically optimal RRT via rewiring
• RRT-Connect — bidirectional growth; much faster in practice
• Informed RRT\* and BIT\* — focusing sampling once a solution exists
• Sampling strategies — goal bias, Gaussian sampling, bridge tests for narrow passages
• Path smoothing and shortcutting — sampling-based paths are jerky and need post-processing
• OMPL — the standard planning library and its algorithm zoo

9.3 Motion Planning with Dynamics

• Kinodynamic planning — planning in state space with velocity and acceleration limits
• Nonholonomic planning — respecting constraints like "cannot move sideways"
• Dubins and Reeds-Shepp paths — shortest paths for car-like vehicles, forward-only and with reversing
• Trajectory optimization — CHOMP, STOMP, TrajOpt; deform an initial guess into a good trajectory
• Direct collocation and shooting methods — the two standard transcriptions of an optimal control problem
• Contact-implicit trajectory optimization — planning through making and breaking contact
• Local planners and reactive control — DWA, Timed Elastic Band, MPPI for obstacle avoidance in the loop
• Velocity obstacles and ORCA — reciprocal collision avoidance among multiple moving agents
• Global-local planner architecture — the standard two-layer navigation design

9.4 Task and Higher-Level Planning

• Symbolic planning — STRIPS and PDDL; reasoning about discrete actions and their preconditions
• Task and Motion Planning (TAMP) — coupling discrete task choices with continuous motion feasibility
• Behaviour trees — the modern standard for robot task orchestration
• Finite state machines — simpler, and adequate for many systems; SMACH and successors
• Hierarchical planning — decomposing a goal into subgoals
• Planning under uncertainty — MDPs and POMDPs; belief-space planning
• Replanning and execution monitoring — detecting that the plan is failing and doing something about it
• Multi-robot coordination — task allocation, conflict-based search, traffic management in warehouses

10.1 ROS 2 Core Concepts

• What ROS is and isn't — a middleware and toolset, not an operating system and not a framework you're locked into
• Why ROS 2 exists — real-time support, multi-robot, security, and production readiness that ROS 1 lacked
• Nodes — the unit of computation; one process or one component
• Topics — anonymous asynchronous publish/subscribe; the primary data flow mechanism
• Messages and interface definitions — `.msg`, `.srv`, `.action` files and code generation
• Services — synchronous request/response for quick queries
• Actions — long-running goals with feedback and cancellation; the right choice for motion commands
• Parameters — runtime configuration, declared and typed; parameter callbacks and validation
• Launch files — Python, XML and YAML launch; composing a system from many nodes
• Namespaces and remapping — running multiple instances without name collisions
• Lifecycle (managed) nodes — explicit configure/activate/deactivate states for deterministic startup
• Executors and callback groups — single-threaded vs multi-threaded, reentrant vs mutually exclusive
• Composition — running multiple nodes in one process for zero-copy intra-process communication
• rclcpp and rclpy — the C++ and Python client libraries, and when the performance difference matters
• Timers, rates and spinning — controlling execution frequency correctly
• Time in ROS 2 — system time, steady time, and simulated time via `/clock` and `use_sim_time`

10.2 ROS 2 Middleware and Quality of Service

• DDS — the underlying middleware standard ROS 2 is built on
• RMW implementations — Fast DDS, Cyclone DDS, Connext; swapping them and why you might
• Discovery — how nodes find each other, and why it becomes a problem at scale
• QoS profiles — reliability, durability, history, depth
• Reliable vs best effort — TCP-like guarantees vs UDP-like speed; sensor data usually wants best effort
• Transient local durability — late-joining subscribers receiving the last message; how latched topics work now
• QoS incompatibility — the silent failure where a publisher and subscriber never connect and nothing errors loudly
• Deadline, liveliness and lifespan — the QoS policies for detecting a dead publisher
• Zero-copy and shared memory transport — for large messages like images and point clouds
• DDS domains and partitions — isolating multiple robots on one network
• ROS 2 security (SROS 2) — authentication, encryption and access control

10.3 ROS 2 Ecosystem and Tooling

• Workspaces and overlays — underlay/overlay, and how sourcing actually works
• colcon — building a workspace; `--symlink-install`, `--packages-select`, parallel builds
• ament — the build system and its CMake and Python variants
• package.xml and dependencies — declaring what your package needs
• rosdep — resolving system dependencies across distributions
• ROS 2 distributions — the annual release cadence, LTS versus non-LTS, and choosing one deliberately
• ros2 CLI — `topic`, `node`, `service`, `param`, `bag`, `doctor`, `interface`; the daily debugging toolkit
• rqt — graph visualization, plotting, image viewing, console
• RViz2 — 3D visualization of everything; and writing custom displays
• rosbag2 — recording and replaying; the single most valuable debugging tool in robotics
• tf2 — the transform library; broadcasting, listening, buffering, and time-travel lookups
• tf2 debugging — `view_frames`, `tf_echo`, and diagnosing extrapolation errors
• URDF and xacro — describing a robot's kinematics, visuals and collision geometry
• SDF — the richer format used by Gazebo
• robot_state_publisher and joint_state_publisher — turning joint values into a tf tree
• ros2_control — the hardware abstraction and controller manager framework
• Controllers and hardware interfaces — writing a controller and a hardware component
• MoveIt 2 — motion planning, kinematics, collision checking and execution for manipulators
• Nav2 — the navigation stack; behaviour trees, costmaps, planners, controllers, recoveries
• micro-ROS — ROS 2 on microcontrollers
• ROS 1 bridge — interoperating with legacy systems
• Diagnostics and monitoring — `diagnostic_updater`, aggregators, and system health reporting

10.4 Real-Time Systems

• What real-time actually means — deterministic deadlines, not raw speed
• Hard, firm and soft real-time — and which parts of a robot need which
• Latency vs jitter — jitter is usually the thing that destroys control performance
• Control loop rates — why current loops run at tens of kHz and planners at a few Hz
• RT_PREEMPT Linux — the standard route to soft/firm real-time on a general-purpose OS
• Real-time operating systems — FreeRTOS, Zephyr, QNX, VxWorks
• Priority inversion and priority inheritance — the classic real-time failure and its fix
• Memory allocation in real-time code — why `malloc` in a control loop is a defect
• Lock-free data structures — passing data between threads without blocking
• CPU isolation and affinity — pinning a control thread away from everything else
• Worst-case execution time (WCET) — the number that matters for certification
• Watchdogs — detecting a hung control loop and failing safe

10.5 Simulation

• Why simulate — cheaper, faster, safer, and repeatable; and where it lies to you
• Gazebo / Gazebo Sim — the ROS-native simulator, with sensor and plugin ecosystems
• MuJoCo — fast and accurate contact dynamics; the research standard for control and RL
• NVIDIA Isaac Sim and Isaac Lab — photorealistic rendering plus GPU-parallel physics for RL at scale
• PyBullet — lightweight, scriptable, widely used in research
• Webots and CoppeliaSim — full-featured alternatives with good education support
• Drake — rigorous multibody dynamics and optimization-based control
• Physics engine choice — contact model, solver, timestep, and the stability implications
• The reality gap — where simulation and the real world diverge, and why contact and friction are worst
• Sensor simulation — camera, LiDAR, IMU models, and their noise
• Domain randomization — deliberately varying simulation parameters so policies transfer
• Software-in-the-loop and hardware-in-the-loop — testing real code and real hardware against a simulated world
• Digital twins — a live simulation mirroring a deployed system

10.6 Software Engineering for Robotics

• C++ for robotics — modern C++, RAII, smart pointers, templates; still the language of real-time code
• Python for robotics — prototyping, tooling, and scripting; and knowing when it's too slow
• Version control for large repos — monorepos, submodules, git-lfs for large assets
• Build systems — CMake proficiency is unavoidable
• Dependency and environment management — Docker containers for reproducible robot software
• Unit testing — gtest, pytest; testing algorithms in isolation
• Integration testing — launch_testing, testing whole node graphs
• Simulation-based CI — running scenario tests on every commit
• Logging — structured logging, log levels, and log volume management on a robot with limited storage
• Configuration management — YAML sprawl, and keeping configuration versioned with the code
• Code review and static analysis — clang-tidy, cppcheck, linters
• Profiling — perf, valgrind, tracing tools; finding the node that's blowing the cycle budget
• Deterministic replay — reproducing a field failure from a bag file

11.1 Microcontrollers and Firmware

• Microcontroller architectures — ARM Cortex-M, ESP32, AVR; picking for the job
• Registers and memory-mapped I/O — talking to hardware directly
• Interrupts and ISRs — priorities, latency, and what you must never do inside one
• DMA — moving data without the CPU; essential for high-rate sensor sampling
• Timers and counters — PWM generation, input capture, encoder quadrature decoding
• Clock configuration — PLLs, prescalers, and the source of many first-day mysteries
• Bare metal vs RTOS — when a superloop is enough and when it isn't
• FreeRTOS / Zephyr — tasks, queues, semaphores, mutexes
• Bootloaders and firmware update — field updates without bricking the robot
• Debugging embedded — JTAG/SWD, printf debugging, logic analysers, oscilloscopes
• Fixed-point arithmetic — control maths without an FPU
• Flash and EEPROM wear — persistent storage of calibration and state
• Brown-out and power-fail handling — behaving safely when the supply dips

11.2 Communication Buses

• UART / serial — the simplest link, and still everywhere
• SPI — fast, synchronous, short-range; sensors and displays
• I2C — multi-device, two wires, slow; addressing conflicts and bus lockups
• CAN bus — differential, robust, arbitration by priority; the automotive and industrial standard
• CANopen — the higher-level protocol layered on CAN for motor drives and I/O
• EtherCAT — deterministic real-time Ethernet; the standard for high-performance multi-axis motion
• Ethernet and UDP/TCP — the general-purpose robot backbone
• Time-sensitive networking (TSN) — deterministic Ethernet without a specialist protocol
• RS-232, RS-422, RS-485 — the serial standards still ubiquitous in industry
• USB — convenient, and a poor choice for anything that must never disconnect
• Wireless: Wi-Fi, Bluetooth, LoRa, 5G — bandwidth, latency and reliability tradeoffs
• Protocol selection — determinism, bandwidth, distance, connector cost, and noise immunity

12.1 PLC Fundamentals

• What a PLC is — a ruggedized deterministic controller built for decades of continuous operation
• Why PLCs rather than a PC — determinism, reliability, environmental tolerance, and certification
• The scan cycle — read inputs, execute program, write outputs, housekeeping; repeat forever
• Scan time — and why an unbounded loop in a PLC program is a serious fault
• Digital I/O — sourcing vs sinking, 24 V logic, wetting current
• Analogue I/O — 4–20 mA and 0–10 V, scaling, and why 4–20 mA detects a broken wire
• I/O modules and racks — local and remote I/O, hot swap
• Tags and addressing — symbolic versus absolute addressing
• Data types — BOOL, INT, DINT, REAL, STRING, and vendor-specific structures
• Memory areas — inputs, outputs, markers, retentive memory, data blocks
• Timers and counters — TON, TOF, TP, CTU, CTD; the fundamental sequencing building blocks
• Latching and sealing circuits — the ladder logic idioms every plant floor uses
• First scan and initialization — establishing a known state at power-up
• Retentive vs non-retentive memory — what survives a power cycle, and what must not

12.2 IEC 61131-3 Programming

• The IEC 61131-3 standard — the five languages and why portability is still imperfect
• Ladder Diagram (LD) — relay-logic notation; universally understood by maintenance electricians
• Function Block Diagram (FBD) — signal-flow notation, good for continuous process control
• Structured Text (ST) — Pascal-like textual programming; the right choice for algorithms and maths
• Sequential Function Chart (SFC) — steps and transitions; ideal for sequential machine operation
• Instruction List (IL) — deprecated, still found in legacy code
• Program organization units — programs, function blocks, and functions
• Function blocks and instances — reusable stateful logic; the closest thing to objects
• Tasks and priorities — cyclic, event-driven and freewheeling tasks
• Structured programming in PLCs — modularity, naming conventions, and avoiding one 5,000-rung routine
• IEC 61131-3 object-oriented extensions — classes, interfaces and inheritance in modern platforms
• PLCopen motion function blocks — the standardized interface for coordinated motion

12.3 Platforms and Vendors

• Siemens — S7-1200/1500, TIA Portal; dominant in Europe
• Rockwell / Allen-Bradley — ControlLogix and CompactLogix, Studio 5000; dominant in North America
• Beckhoff and CODESYS — PC-based control, TwinCAT; the bridge between IT and OT
• Mitsubishi, Omron, Schneider, ABB — significant regional and sector presence
• Soft PLCs — control running on standard hardware with a real-time kernel
• Vendor lock-in — the practical reality of the industry, and how it shapes projects
• Licensing and toolchain cost — a genuine barrier to entry, and worth planning for

12.4 Industrial Networks

• Fieldbus vs industrial Ethernet — the generational split
• Modbus RTU and Modbus TCP — simple, ancient, universally supported
• PROFIBUS — the legacy Siemens fieldbus
• PROFINET — Ethernet-based, with real-time and isochronous classes
• EtherNet/IP — the Rockwell-aligned Ethernet protocol, built on CIP
• EtherCAT — sub-millisecond cycle times and precise synchronization
• CC-Link IE, POWERLINK, SERCOS — the other significant industrial Ethernet families
• IO-Link — point-to-point sensor and actuator communication below the fieldbus layer
• OPC UA — vendor-neutral information modelling and secure data exchange; the OT/IT bridge
• MQTT and Sparkplug B — lightweight publish/subscribe for industrial telemetry
• Network topology and determinism — star, ring, line; redundancy protocols like MRP and PRP
• Industrial network diagnostics — the tooling for finding a marginal connector on a live line

12.5 HMI, SCADA and Data

• HMI design — operator interfaces; the high-performance HMI philosophy of grey screens and meaningful colour
• Alarm management — rationalization, prioritization, and avoiding alarm floods (ISA-18.2)
• SCADA systems — supervisory control and data acquisition across a plant
• Historians — time-series storage of process data at scale
• Recipe and batch management — parameterized production; ISA-88
• MES and ERP integration — where the plant floor meets the business systems; ISA-95 levels
• Industry 4.0 and IIoT — edge gateways, cloud connectivity, and the security implications
• OT cybersecurity — IEC 62443, network segmentation, the Purdue model, and why air gaps mostly aren't

12.6 Industrial Robot Programming

• Teach pendant programming — jogging, teaching points, and the workflow most integrators actually use
• Vendor languages — KRL (KUKA), RAPID (ABB), Karel and TP (FANUC), URScript (Universal Robots)
• Online vs offline programming — teaching on the robot versus simulating and downloading
• Offline programming and simulation tools — RoboDK, Process Simulate, RobotStudio, Delmia
• Tool and work object calibration — TCP calibration by multi-point touch-up
• Coordinate systems on industrial robots — world, base, tool, work object frames
• Motion commands — joint, linear, circular; blending and zone parameters
• I/O and PLC integration — the robot as one device in a larger cell
• Program structure and error handling — recovering from a part not present or a gripper failure
• Cycle time optimization — the metric the customer actually cares about
• Robot cell design — layout, reach, fixturing, part presentation, and singularity avoidance
• Machine tending, palletizing, welding, dispensing — the four applications that dominate installed base

12.7 Safety and Functional Safety

• Risk assessment — ISO 12100; the process everything else follows from
• Hazard identification — pinch points, crush, impact, entanglement, stored energy
• The hierarchy of controls — eliminate, substitute, engineer, administrate, PPE; in that order
• ISO 13849-1 and Performance Level — PLr a through e, categories B/1/2/3/4, MTTFd, DC, CCF
• IEC 62061 and SIL — the alternative safety integrity framework
• IEC 61508 — the parent functional safety standard
• ISO 10218-1 and -2 — safety requirements for industrial robots and for their integration
• ISO/TS 15066 — collaborative robot operation; force and pressure limits by body region
• The four collaborative modes — safety-rated monitored stop, hand guiding, speed and separation monitoring, power and force limiting
• Emergency stop — categories 0, 1 and 2; E-stop is not a safeguard, it's a last resort
• Safety relays and safety PLCs — dual-channel architecture, cross-monitoring, diagnostic coverage
• Light curtains and area scanners — safety distance calculation from approach speed and response time
• Interlocked guards — and the defeat-resistance requirements
• Two-hand control and enabling devices — three-position enabling switches
• Safe torque off (STO) and safe motion functions — SS1, SS2, SLS, SLP at the drive level
• Validation and verification — proving the safety function actually works, and documenting it
• Machinery Directive / Regulation and CE marking — the European legal framework
• OSHA and ANSI/RIA R15.06 — the North American equivalents
• Functional safety is not cybersecurity — related, increasingly coupled, and distinct disciplines

13.1 Grasping

• The grasping problem — choosing where and how to make contact so the object stays held
• Force closure — the grasp can resist any external wrench using friction
• Form closure — geometry alone constrains the object, no friction needed
• Grasp quality metrics — epsilon quality, wrench space volume, and their limitations
• Antipodal grasps — two opposing contacts within the friction cone; the basis of most parallel-jaw grasping
• Analytical grasp synthesis — computing grasps from a known object model
• Data-driven grasp detection — Dex-Net, GraspNet, GG-CNN; predicting grasps directly from images or point clouds
• Grasp pose detection from partial views — the realistic case, where you never see the whole object
• Bin picking — cluttered, occluded, unknown pose; the canonical hard industrial vision task
• Singulation — separating one item from a pile before grasping it
• Suction grasping — often more practical than fingers; surface quality and porosity determine feasibility
• Grasp execution and failure recovery — detecting a failed grasp and retrying sensibly

13.2 End Effectors

• Parallel-jaw grippers — simple, robust, and adequate for a surprising proportion of tasks
• Vacuum and suction cups — the workhorse of logistics and packaging
• Magnetic grippers — for ferrous parts
• Multi-fingered hands — dexterous, expensive, and hard to control
• Underactuated and adaptive grippers — fewer motors than joints, conforming passively to shape
• Soft grippers — compliant materials handling fragile or irregular objects
• Tool changers — automatic end-effector swapping for multi-task cells
• Gripper force control — holding firmly enough not to drop and gently enough not to crush
• Custom fixturing and tooling — often a better answer than a cleverer gripper

13.3 Manipulation Beyond Pick and Place

• Contact-rich manipulation — insertion, assembly, connector mating; where models are least reliable
• Peg-in-hole and search strategies — spiral search, compliant insertion, force-guided alignment
• Remote centre of compliance (RCC) — passive mechanical assistance for assembly
• In-hand manipulation — repositioning an object without releasing it
• Non-prehensile manipulation — pushing, sliding, toppling; manipulating without grasping
• Deformable object manipulation — cloth, cable, food; state is high-dimensional and dynamics are hard
• Bimanual manipulation — two arms, coordinated constraints, closed kinematic chains
• Mobile manipulation — an arm on a base, where base positioning becomes part of the manipulation problem
• Manipulation planning — planning through contact mode changes and regrasps
• Tool use — grasping something in order to act on something else

14.1 Platforms

• Wheeled robots — the efficient default on flat ground
• Tracked robots — traction on rough terrain, at the cost of odometry and turning efficiency
• Legged robots — quadrupeds and bipeds; capability on unstructured terrain, enormous control complexity
• Aerial robots — multirotors and fixed wing; freedom of motion versus endurance
• Marine and underwater — buoyancy, currents, and the absence of GPS or radio underwater
• Platform selection — terrain, payload, endurance, cost, and regulatory constraints

14.2 Navigation

• The navigation stack architecture — localization, global planner, local planner, controller, recovery behaviours
• Nav2 — the ROS 2 navigation framework and its behaviour-tree orchestration
• Costmaps — occupancy, inflation layers, obstacle layers, static layers, and layered costmap composition
• Inflation radius and robot footprint — the parameters that decide whether the robot fits through the door
• Global planners — NavFn, Smac Planner, Theta\*
• Local controllers — DWB, TEB, Regulated Pure Pursuit, MPPI
• Recovery behaviours — clearing costmaps, rotating in place, backing up; what happens when stuck
• Waypoint following and route graphs — structured navigation in known environments
• Docking and precision alignment — charging contacts and conveyor handoff need millimetre accuracy
• Dynamic obstacle handling — people move, and the map doesn't know
• Social navigation — behaving predictably and legibly around humans
• Multi-floor navigation — lifts, maps per floor, and transitions between them
• GNSS/GPS and RTK — outdoor absolute positioning, and centimetre accuracy with corrections
• GNSS-denied navigation — indoors, underground, and under jamming

14.3 Legged Locomotion

• Why legs are hard — underactuated, hybrid dynamics, and a small support polygon
• Static vs dynamic stability — standing versus falling forwards in a controlled way
• Zero Moment Point (ZMP) — the classical criterion for balance
• Capture point and divergent component of motion — where to step to stop
• Centre of pressure and support polygon — the physical basis of balance
• Gaits — walk, trot, pace, bound, gallop; and gait transitions
• Simplified models — linear inverted pendulum, spring-loaded inverted pendulum, centroidal dynamics
• Footstep planning — where to place feet on uneven or discrete terrain
• Whole-body control for legged robots — hierarchical QP with contact constraints
• MPC for locomotion — the current standard for dynamic legged control
• RL for locomotion — trained in massively parallel simulation, transferred with domain randomization; now genuinely dominant
• Terrain perception for locomotion — elevation mapping and traversability estimation
• Fall detection and recovery — getting back up is a real capability requirement

14.4 Aerial Robotics

• Multirotor dynamics — underactuated; four inputs, six degrees of freedom
• Thrust and torque mixing — mapping desired body wrench to individual motor commands
• Cascaded control architecture — position outer loop, attitude inner loop, rate innermost
• Attitude estimation — the complementary or EKF filter fusing IMU and magnetometer
• Differential flatness — why multirotor trajectory generation is tractable
• Minimum snap trajectory generation — the standard method for aggressive flight
• PX4 and ArduPilot — the open source autopilot stacks
• MAVLink — the communication protocol between autopilot and companion computer
• Fixed-wing and VTOL — endurance and the transition control problem
• Failsafes — return to launch, geofencing, motor failure handling
• Airspace regulation — BVLOS, remote ID, and the operational limits that actually govern deployment

14.5 Fleets and Warehouse Robotics

• AGVs vs AMRs — fixed-path guided vehicles versus autonomously navigating robots
• Fleet management systems — task allocation, traffic control, deadlock avoidance
• Charging strategy — opportunity charging, battery swap, and duty cycle planning
• VDA 5050 — the standard interface between fleet managers and vehicles from different vendors
• Warehouse execution and WMS integration — the robot is one part of a much larger system
• Throughput modelling — simulating a fleet to size it before buying anything
• Mixed human-robot environments — the dominant real deployment condition

15.1 Framing

• What "physical AI" means — learned systems that perceive and act in the physical world under real-time and safety constraints
• Why robotics is harder than other ML domains — data is expensive, mistakes have physical cost, and the model's actions change its own data distribution
• The data bottleneck — there is no internet-scale corpus of robot interaction, and this is the field's central problem
• Classical vs learned components — a modular pipeline with learned perception, or end-to-end; and the honest tradeoffs
• Where learning genuinely wins — perception, contact-rich skills, and generalization to object variation
• Where classical methods still win — anything with a good model, anything safety-critical, anything needing guarantees

15.2 Imitation Learning

• Behaviour cloning — supervised learning on state-action pairs from demonstrations
• Compounding error and covariate shift — the fundamental flaw of naive behaviour cloning
• DAgger — iteratively collecting expert corrections on the policy's own state distribution
• Multimodality in demonstrations — humans do the same task different ways, and averaging them produces nonsense
• Action chunking (ACT) — predicting a sequence of actions at once to reduce compounding error
• Diffusion policy — modelling the action distribution with a diffusion model; handles multimodality well
• Inverse reinforcement learning — inferring the reward function from demonstrations
• Goal-conditioned imitation — one policy, many tasks, specified at inference
• How many demonstrations — the practical question, and how sharply it varies by task

15.3 Reinforcement Learning for Robotics

• The RL formulation — states, actions, rewards, policies, value functions
• Model-free algorithms — PPO, SAC, TD3; what's actually used on robots
• Model-based RL — learning dynamics and planning within them; far more sample efficient
• Sample efficiency — the binding constraint on any real-robot RL
• Reward shaping — and reward hacking, where the agent optimizes exactly what you wrote
• Sparse rewards and exploration — curriculum learning, hindsight experience replay
• Safe RL — constrained MDPs, shielding, and control barrier functions as a safety filter
• Offline RL — learning from logged data without further interaction
• Massively parallel simulation — Isaac Lab and thousands of simultaneous environments on one GPU
• Residual RL — learning a correction on top of a classical controller rather than replacing it

15.4 Sim-to-Real

• The reality gap — every mismatch between simulator and world, and which ones matter
• Domain randomization — randomizing dynamics, appearance, latency and noise so the policy can't overfit to the sim
• System identification — measuring your real robot's parameters to make the simulator more accurate
• Real-to-sim — building simulation assets from real scans
• Actuator modelling — usually the largest single source of sim-to-real gap
• Latency modelling — real sensors and actuators have delays that simulators often omit
• Observation and action space design — choosing representations that transfer
• Fine-tuning on real data — a small amount of real interaction after large-scale simulated training
• When sim-to-real fails — contact-rich tasks, deformables, and anything where friction matters

15.5 Foundation Models for Robotics

• Vision-language-action (VLA) models — a single model mapping images and instructions to actions
• The lineage — RT-1, RT-2, OpenVLA, Octo, π0 and successors; a fast-moving area
• Cross-embodiment learning — training across different robot bodies to share data; the Open X-Embodiment effort
• Pretrained vision encoders for robotics — using DINOv2, SAM or CLIP features rather than training from pixels
• LLMs for task planning — decomposing an instruction into steps; SayCan and successors
• Code generation as a robot interface — an LLM writing the policy or the plan rather than the actions
• Grounding language in the physical world — the hard part; a model that talks fluently about physics may not obey it
• Evaluation of generalist policies — genuinely unsolved; benchmarks are immature and real-world evaluation is expensive
• Latency and deployment constraints — a large model in a 10 Hz control loop is a systems engineering problem
• Honest assessment — impressive demonstrations, limited reliability, and a large gap between video and deployment

15.6 World Models and Prediction

• Learned dynamics models — predicting the next state given state and action
• Latent dynamics — Dreamer-style models that predict in a compressed space
• Video prediction models — predicting future frames as an implicit world model
• Planning inside a learned model — MPC with a neural dynamics model
• Model error compounding — why long-horizon rollouts in learned models degrade
• Uncertainty-aware models — ensembles and probabilistic dynamics for knowing when not to trust the model

15.7 Data Collection and Teleoperation

• Teleoperation interfaces — VR controllers, leader-follower arms, exoskeletons, space mice
• Low-cost teleop rigs — ALOHA, GELLO, and the shift toward affordable data collection
• Handheld data collection — UMI-style grippers that collect demonstrations without a robot present
• Kinesthetic teaching — physically guiding a backdrivable arm through the motion
• Data quality versus quantity — a small set of consistent demonstrations often beats a large inconsistent one
• Dataset formats and tooling — LeRobot, RLDS, and the standardization effort
• Open datasets — Open X-Embodiment, DROID, BridgeData
• Autonomous data collection — self-supervised practice, and the safety problem it creates
• Scaling laws for robot data — an open question, and the field's most consequential one

• Collaborative robots — arms designed to share a workspace with people; power and force limiting
• Speed and separation monitoring — dynamic safety zones that shrink the robot's speed as a person approaches
• Human detection and tracking — the perception requirement underlying every safety function
• Intent prediction — anticipating where a person is going to reach
• Legible motion — moving so a human can correctly infer what the robot will do next
• Shared autonomy — blending human input with autonomous assistance
• Handover — passing an object to or from a person; deceptively difficult
• Interfaces — voice, gesture, touchscreen, AR overlays; and matching interface to task
• Trust and over-trust — both under-reliance and complacency are failure modes
• Mental models — what the operator believes the robot is doing versus what it is doing
• Anthropomorphism — the expectations a humanoid form creates and usually fails to meet
• Workplace acceptance — the deployment factor most technical teams underestimate

17.1 Testing

• Unit and integration testing — as in any software, plus hardware mocks
• Simulation-based regression testing — running scenario suites in CI
• Hardware-in-the-loop testing — real controllers against simulated plant
• Scenario-based testing — enumerating the situations the robot must handle
• Fault injection — deliberately failing sensors, networks and actuators to test degradation
• Long-duration soak testing — the failures that only appear after 200 hours
• Edge case and adversarial testing — reflective floors, glass walls, sunlight through a window
• Acceptance testing — the criteria that determine whether the customer signs

17.2 Reliability and Maintenance

• MTBF and MTTR — the two numbers a customer's operations team actually cares about
• FMEA — failure modes and effects analysis; systematic enumeration of what can go wrong
• Fault tree analysis — reasoning backwards from a hazard to its causes
• Graceful degradation — continuing to operate safely with reduced capability
• Redundancy — where it's worth the weight and cost, and where it just adds failure modes
• Predictive maintenance — vibration and current signature analysis to catch bearing wear early
• Spares and serviceability — designing for field replacement by a non-expert
• Wear items — belts, bearings, cables, gripper pads; and planning their replacement cycle

17.3 Deployment and Operations

• Site survey and commissioning — the environment is never what the drawings said
• Calibration in the field — and re-calibration after a collision
• Fleet software updates — staged rollout, rollback, and never bricking a robot remotely
• Remote monitoring and telemetry — knowing a robot is degrading before the customer calls
• Remote diagnostics and teleassist — a human resolving the cases autonomy can't
• Data pipelines from the field — logging enough to debug without saturating the network
• Incident investigation — bag replay, root cause analysis, and corrective action
• Operator training and documentation — a substantial deliverable, routinely underestimated
• Total cost of ownership — what the customer is really evaluating

• C++ — the language of real-time robotics; modern C++ and its idioms
• Python — prototyping, scripting, ML, and the whole tooling layer
• Rust — growing interest for safety-critical embedded and middleware
• MATLAB/Simulink — still dominant in control design and automotive
• Eigen — the C++ linear algebra library everything is built on
• OpenCV — classical computer vision
• PCL and Open3D — point cloud processing
• Ceres, g2o, GTSAM — nonlinear optimization and factor graphs
• Pinocchio — fast rigid body dynamics with analytical derivatives
• Drake — multibody dynamics, optimization, and verification
• OMPL — sampling-based motion planning
• CasADi — symbolic framework for optimal control and NMPC
• PyTorch / JAX — the learning side
• NVIDIA Jetson — the standard edge compute platform for robots
• Compute selection — CPU, GPU, FPGA, and dedicated accelerators; matching hardware to workload
• Common research platforms — UR arms, Franka, Kinova, Unitree, TurtleBot, and what each is good for teaching
• The vendor landscape — industrial arms, cobots, AMRs, and the integrators who deploy them

19.1 Building Competence

• Build something that moves — a line follower, a balancing robot, a small arm; the physical debugging skill only comes from hardware
• Simulate first, then break it in reality — and learn exactly which assumptions failed
• Reproduce a paper — the fastest route from reading to understanding
• Work with a real industrial robot — even briefly; it recalibrates expectations enormously
• Write a PID controller from scratch — and tune it on real hardware with real friction
• Implement a Kalman filter from scratch — before ever using a library one
• Do a full calibration — camera intrinsics, hand-eye, and tool centre point
• Debug with an oscilloscope — some faults are invisible from software
• Take a system from prototype to something that runs unattended for a week — this is where the real learning is

19.2 Staying Current

• Key conferences — ICRA, IROS, RSS, CoRL, and Humanoids
• Reading papers critically — especially distinguishing a demonstration from a capability
• The video-to-deployment gap — an impressive clip may represent one success in fifty attempts
• Open source participation — ROS, MoveIt, Nav2 and the rest are maintained by people you can talk to
• Standards literacy — knowing which standards apply to your sector is a genuine professional differentiator

19.3 Specialization Paths

• Controls engineer — dynamics, control theory, real-time systems
• Perception engineer — computer vision, sensor fusion, deep learning
• Motion planning engineer — algorithms, optimization, computational geometry
• Robotics software engineer — ROS 2, C++, architecture, integration
• Embedded/firmware engineer — microcontrollers, drivers, real-time, hardware bring-up
• Mechanical/mechatronics engineer — design, actuation, structures
• Controls and automation engineer — PLCs, industrial networks, plant floor integration
• Systems integrator — designing and commissioning complete cells and lines
• Robot learning researcher — imitation, RL, foundation models
• Safety engineer — risk assessment, functional safety, certification
• Field/deployment engineer — the role that discovers what the other roles got wrong

19.4 Two Cultures Worth Understanding

• Research robotics vs industrial automation — one optimizes for capability and novelty, the other for uptime and cost per part; both call themselves robotics and they share surprisingly little vocabulary
• Why industrial systems look conservative — a line stopping costs thousands per minute, and a clever solution that fails once a week is worse than a dull one that never does
• Why research systems look fragile — they are demonstrating that something is possible, which is a different objective from demonstrating it is reliable
• The translation problem — moving a capability from one culture to the other is itself a hard engineering discipline, and where much of the current opportunity sits