Autonomous Robotics — Personal Cheatsheet

0 · Trig, vectors, calculus basics

θ	0°	30°	45°	60°	90°
sin	0	1/2	√2/2	√3/2	1
cos	1	√3/2	√2/2	1/2	0
tan	0	1/√3	1	√3	∞

sin(180−θ)=sin θ · cos(180−θ)=−cos θ · sin(−θ)=−sin θ · cos(−θ)=cos θ. E.g. cos 135°=−√2/2, sin(−120°)=−√3/2, cos(−120°)=−½.

● atan2(y,x): angle of (x,y) in the correct quadrant (atan(y/x) is wrong for x<0). Angle of (3,3)=45°; of (−2,−2√3): atan2 gives −120°.

Vector norm ‖v‖=√(v_x²+v_y²+v_z²); unit vector v/‖v‖; dot a·b=‖a‖‖b‖cos θ; cross x̂×ŷ=ẑ (right-hand rule).

Chain rule: d/dt f(s(t)) = f′(s)·ṡ. Product: (fg)′=f′g+fg′. ẋ=dx/dt (rate), ẍ=acceleration.

Small angle: sin θ≈θ, cos θ≈1 (θ in rad).

1 · State-space & motion

✔ Continuous: ẋ=f(x,u), y=h(x,u) · discrete: x_k+1=f_k(x_k,u_k). State x = minimal info making the future independent of the past; u = control input; y = sensor output.

● Constant acceleration over Δt

v_t+1 = v_t + aΔt x_t+1 = x_t + v_tΔt + ½aΔt²

Invert for a: a = 2(x₁−x₀−v₀Δt)/Δt² (unique only if a const.)

● Position accumulates more error than velocity (integration sums errors) → dead reckoning drifts → external sensors needed.

● Trajectory = path X(s) + time scaling s(t): Ẋ=X′(s)ṡ; Ẍ=X″ṡ²+X′s̈ (don't drop the s̈ term).

● Integrators: Euler x+=f·dt (err O(dt²)/step); midpoint: slope at half-step; RK4: 4 slopes weighted 1-2-2-1 (err O(dt⁵)/step).

● Double integrator, state (x,y,v_x,v_y), u=(a_x,a_y): x_t+1=Ax_t+Bu_t, A rows: [1 0 Δt 0; 0 1 0 Δt; 0 0 1 0; 0 0 0 1], B: [½Δt² 0; 0 ½Δt²; Δt 0; 0 Δt].

2 · Wheels & constraints

✔ Holonomic g(q,t)=0 (limits positions); velocity constraint A(q,t)q̇=b (non-holonomic if not integrable → limits velocities, e.g. no sideways motion). Omni robot = holonomic; diff-drive/car = non-holonomic.

✔ No-slip: ẋ sin θ − ẏ cos θ = 0 · rolling: ẋ cos θ + ẏ sin θ = rφ̇.

● Differential drive (r wheel radius, L separation)

v = r(ω_R+ω_L)2 ω = r(ω_R−ω_L)L R_turn = vω

wheel speed v_i=rω_i; equal ω's → straight; opposite → spin in place.

● Unicycle: ẋ=v cos θ, ẏ=v sin θ, θ̇=ω.

● Bicycle (wheelbase L, steering α)

x⁺=x+v cosθΔt · y⁺=y+v sinθΔt · θ⁺=θ+v tan αLΔt

turning radius R=L/tan α. Steering error → orientation error (redirects all later motion); velocity error → distance error.

3 · Lagrangian mechanics

✔ T=½mv², V=mgy, L=T−V; Euler–Lagrange: d/dt(∂L/∂q̇)−∂L/∂q = Q.

✔ Pendulum (mass m, rod l): T=½ml²θ̇², V=mgl(1−cosθ) → ml²θ̈+mgl sinθ=0, θ̈=−(g/l)sinθ.

● With viscous friction b: add Q=−bθ̇ (non-conservative → right side, NOT into V):

ml²θ̈ + bθ̇ + mgl sin θ = 0

(Ex 7: m=2, l=1, b=0.1, g=10 → 2θ̈+0.1θ̇+20 sinθ=0.) I=ml² (point mass). Alternative: torque balance Iθ̈=−mgl sinθ−bθ̇.

4 · Rotations (SO(3))

● R∈SO(3): RᵀR=I, det R=1, R⁻¹=Rᵀ. Columns of R_sa = axes of {a} in {s} coords. Build 3rd axis: ẑ=x̂×ŷ.

● Subscript cancellation: R_abR_bc=R_ac; R_abp_b=p_a; R_ab=R_asR_sb=R_saᵀR_sb.

● Composition side: R_sb′=R·R_sb = rotate about world-frame axis (premultiply); R_sb″=R_sb·R = rotate about body-frame axis (postmultiply). Fixed-frame sequence ⇒ stack left: last applied is leftmost.

● Elementary rotations (✔ only R_y is on the sheet)

R_x(γ)=1 0 0
0 cosγ −sinγ
0 sinγ cosγ R_y(β)=cosβ 0 sinβ
0 1 0
−sinβ 0 cosβ

R_z(α)=cosα −sinα 0
sinα cosα 0
0 0 1 pattern: 1 on the axis; −sin above +sin except R_y (flipped)

ZYX Euler (α,β,γ) body-frame: R = R_z(α)R_y(β)R_x(γ).

● Skew matrix of ω=(ω₁,ω₂,ω₃): [ω] = rows [0 −ω₃ ω₂; ω₃ 0 −ω₁; −ω₂ ω₁ 0]; [ω]ᵀ=−[ω]; [ω]x=ω×x.

● Axis-angle/exp coords ω̂θ (‖ω̂‖=1): Rodrigues R=e^[ω̂]θ=I+sinθ[ω̂]+(1−cosθ)[ω̂]²; angular velocity ω=ω̂θ̇; ṘR⁻¹=[ω_s], R⁻¹Ṙ=[ω_b].

2D rotation: R(θ)=[cosθ −sinθ; sinθ cosθ]; circular motion about C: P(θ)=C+R(θ)r_CP.

5 · Rigid transforms

✔ ^cP=^cR_W^WP+^ct_W; homogeneous ^cT_W=[R t; 0 0 0 1], ^cP̃=^cT_W^WP̃ (append 1 to the point).

✔ Inverse — translation is NOT −t:

T⁻¹ = [Rᵀ −Rᵀt; 0 0 0 1] ⇔ P_W=Rᵀ(P_C−t)

Planar pose: T=[cosθ −sinθ x; sinθ cosθ y; 0 0 1]. Camera center in world = −Rᵀt.

6 · Camera projection

✔ Pipeline: P_W →(R,t)→ P_C →(÷Z_C)→ (x_n,y_n) →(K)→ (u,v). x_n=X_C/Z_C, y_n=Y_C/Z_C; u=f_xx_n+c_x, v=f_yy_n+c_y; K=[f_x s c_x; 0 f_y c_y; 0 0 1]; λp̃=K[R|t]P̃_W; normalize (ũ,ṽ,w̃)↦(ũ/w̃, ṽ/w̃).

● Back-projection: x_n=(u−c_x)/f_x, y_n=(v−c_y)/f_y; ray d=(x_n,y_n,1); with known depth: X_C=x_nZ_C, Y_C=y_nZ_C. Pixel ↦ ray, not point (depth lost).

Visible ⇔ Z_C>0 (Z=0 unprojectable). Metric image plane: x=f·X_C/Z_C (f in length units). Offsets scale like 1/Z_C.

Intrinsic=internal geometry (f_x,f_y,c_x,c_y,skew; in K; fixed by hardware). Extrinsic=camera pose (R,t; changes with motion). Principal point=(c_x,c_y) where optical axis hits image; resolution≈2·(c_x,c_y). Skew≠0 only for non-perpendicular pixel axes. Barrel distortion: lines bulge outward (magnification ↓ with radius, wide-angle); pincushion: inward (telephoto). Homogeneous coords make translation+rotation+projection all matrix products.

7 · Ceiling markers (course setup)

✔ Camera looks up, marker height h → all points share Z_C=h: s = f/h [px/m]; Δu = sΔX; ΔX = Δu/s; h doubles → s halves.

✔ Heading: θ_r = θ_m − θ_i (marker's true − observed orientation). Sheet page 2!

● Pixel noise → metric: ΔX_err=Δu_err/s. One symmetric marker fixes position not heading → need 2 markers (or asymmetric marker). Ceiling: fixed, unoccluded, rarely moved. West=−x, North=+y.

8 · Triangulation, SfM, flow, events

● Triangulation: 3D point = intersection of viewing rays from ≥2 known poses; each view: λp̃=PX̃ (P=K[R|t], 3×4) → 2 linear homogeneous eqs → stack: AX̃=0; solve: SVD A=UΣVᵀ, X̃ = last column of V (smallest σ; minimizes ‖AX̃‖ on ‖X̃‖=1).

● Reprojection error ‖p−p̂‖₂. Baseline ↑ → depth accuracy ↑ but matching harder. Stereo depth: Z=f·b/d (d=disparity).

● SfM = one moving camera, unknown poses; estimates structure+motion jointly (scale unknown). Pipeline: detect → match → pose est. → triangulate → bundle adjustment (joint nonlinear refinement of all poses+points minimizing total reprojection error).

● Optical flow: I_xu+I_yv+I_t=0 (1 eq, 2 unknowns). Assumes brightness constancy (+ small motion). Aperture problem: only flow along the gradient is observable — motion along an edge invisible.

● Event camera: async per-pixel events (x,y,t,p) on log-brightness change (p=polarity ±1) vs sync full frames; µs latency, high dynamic range, no absolute intensity.

9 · Bayes filter · KF · EKF · SLAM

● Bayes' rule (write normalizer separately!)

P(A|z) = P(z|A)P(A)P(z), P(z)=Σ_iP(z|A_i)P(A_i)

check: posteriors sum to 1; belief moves toward sensor-favored hypothesis.

● Bayes filter — predict: bel(x_t)=∫p(x_t|u_t,x_t−1)bel(x_t−1)dx (uncertainty grows); correct: bel(x_t)=η·p(z_t|x_t)bel(x_t) (shrinks).

● p(x_t|u_t,x_t−1)=motion model; p(z_t|x_t)=sensor model; bel(x_t)=belief (posterior over states given all data).

● Kalman filter (linear + Gaussian)

predict: μ̄=Aμ+Bu; Σ̄=AΣAᵀ+R

gain: K=Σ̄Cᵀ(CΣ̄Cᵀ+Q)⁻¹

update: μ⁺=μ̄+K(z−Cμ̄); Σ⁺=(I−KC)Σ̄

innovation=z−Cμ̄ (measured−expected); R=process noise, Q=measurement noise. Gaussian: fully described by (μ,Σ) and closed under linear+Gaussian ops → efficient exact updates.

● EKF: nonlinear f,h → linearize at estimate: A=∂f/∂x, C=∂h/∂x (Jacobians). Use when uncertainty ≈ Gaussian + good initial guess; particle filter for global/multi-modal localization.

● SLAM: chicken-and-egg — localization needs map, mapping needs pose → estimate jointly. EKF-SLAM state: (x, y, θ, m_1x, m_1y, m_2x, m_2y, …). Data association = which measurement ↔ which landmark; wrong association corrupts filter.

● Range → circle (2D) / sphere (3D) around landmark (trilateration d=v_soundΔt); ≥2 identified landmarks → unique pose. Identical rooms → multi-modal belief. Known quantities (e.g. compass heading) are inputs, not state.

10 · Planning & control

✔ Open-loop u=u_planned(t) (no sensing, drifts) · closed-loop policy u=π(x) · feedback u=K(x_ref−x).

● Dijkstra: f(n)=g(n) · A*: f(n)=g(n)+h(n) g=cost from start, h=estimated cost to goal. Admissible: h never overestimates → A* optimal. Consistent: triangle inequality → no reopening. h≡0 ⇒ A*=Dijkstra. Good h → fewer expansions, goal-directed.

● Open-loop grid localization: belief = set of possible cells; M (drive to wall) collapses corridors to end cells; with step counting, each M's count is a measurement pruning hypotheses (predict+correct).

11 · FSM · RL · ROS (definitions)

● FSM: states=behaviors, labeled transitions=events; global interrupt (battery low→Recharge). + simple/interpretable/verifiable; − state explosion, poor under uncertainty.

● RL: state=agent's situation; action=chosen control; reward=scalar feedback per step, maximize cumulative sum; policy π: states→actions, u=π(x) or π(u|x) — the thing RL learns.

● ROS: node=process doing one job; topic=named pub/sub channel; message=typed data on a topic; master (ROS 1)=discovery/name service only — traffic is peer-to-peer.

12 · Sanity checks & traps

Any rotation: det R=1, ‖columns‖=1, RRᵀ=I; rotated vectors keep norm. (Caught a real error in official Solution 4!)
Probabilities sum to 1 · units survive formulas (rad/s·m=m/s) · Z_C≤0 not visible.
Inverse transform translation = −Rᵀt (never just −t, unless R=I).
Sheet's camera formulas assume CAMERA coords — transform world points first.
Sheet's pendulum is frictionless — question with b needs +bθ̇ term.
Ẍ needs BOTH X″ṡ² and X′s̈ terms.
Fixed-axis rotations premultiply (left); body-axis postmultiply (right).
30°/45°/60° answers in surds (sheet p.2), don't round early; store calculator intermediates in A–F.
fx-991CW: SETTINGS→Angle Unit→Degrees; check sin 30=0.5. Matrix app for 3×3 products/Det; Equation app for linear systems.

13 · Worked micro-examples (pattern anchors)

Diff drive r=0.1, L=0.5, ω_L=4, ω_R=6: v=0.1·10/2=0.5 m/s; ω=0.1·2/0.5=0.4 rad/s; R=0.5/0.4=1.25 m.

Projection K=[800 0 320; 0 800 240; 0 0 1], P_C=(1,2,5): x_n=0.2, y_n=0.4 → u=800·0.2+320=480, v=800·0.4+240=560.

Back-projection p=(720,440), same K: x_n=(720−320)/800=0.5, y_n=0.25 → ray (0.5,0.25,1); depth 4 → P_C=(2,1,4).

World→pixel P_W=(3,2,10), t=(−1,0,0), R=I: P_C=(2,2,10) → u=800·0.2+320=480, v=800·0.2+240=400.

Bayes P(A)=0.7, P(z|A)=0.9, P(B)=0.3, P(z|B)=0.2: P(z)=0.63+0.06=0.69 → P(A|z)=0.63/0.69≈0.913, P(B|z)≈0.087.

Ceiling marker f=800, h=4: s=200 px/m; move 0.5 m → 100 px; ±10 px noise → ±10/200=±5 cm; heading 45°−20°=25°.

Pendulum m=2, l=1, b=0.1, g=10: 2θ̈+0.1θ̇+20 sinθ=0 ⇔ θ̈+0.05θ̇+10 sinθ=0.

Optical flow I_x=2, I_y=−1, I_t=−3, u=1: 2−v−3=0 → v=−1 (check: 2+1−3=0 ✓).

Reprojection p=(320,200), p̂=(312,198): ‖(8,2)‖=√68=2√17≈8.25 px.

EKF-SLAM state, 2 landmarks: (x, y, θ, m_1x, m_1y, m_2x, m_2y)ᵀ ∈ ℝ⁷.

Tetrahedron EKF dims: state ℝ¹² (p,a,v,ω); z ∈ ℝ⁸ (4 pts × 2); A 12×12, C 8×12.

Mock test (Moodle "Test") = collection Ex 1–7: Bayes filter, EKF modeling, frames, rotation composition, rigid body, path scaling, pendulum — rehearse those first. Exam 17.07.2026 · allowed: official sheet + fx-991CW.

Autonomous Robotics — exam cheatsheet · SS 2026 · companion to the official formula sheet + fx-991CW