Cheat sheet

Robotics Equations Cheat Sheet

This document summarizes key equations from the provided robotics course slides.

02 Statics

• Newton’s First Law (Linear & Angular):

  • ∑F=0: The sum of forces acting on a body at rest or moving at a constant velocity is zero.

  • ∑τ=0: The sum of torques (moments) acting on a body at rest or rotating at a constant angular velocity is zero.

• Conservation of Momentum:

  • p=m⋅v: Linear momentum (p) is mass (m) times velocity (v).

  • L=I⋅ω: Angular momentum (L) is moment of inertia (I) times angular velocity (ω).

• Newton’s Second Law (Linear & Angular):

  • F=m⋅a: Force (F) equals mass (m) times acceleration (a).

  • τ=I⋅α: Torque (τ) equals moment of inertia (I) times angular acceleration (α).

• Torque from a Force:

  • τ=r×F: Torque (τ) is the cross product of the distance vector (r) from the point of application and the force vector (F).

• Wrench:

  • ω=[F τ​]∈R6: A wrench combines force (F) and torque (τ) into a single 6D vector.

• Coulomb Friction (Dry Friction):

  • Ff​≤μFN​: The friction force (Ff​) is less than or equal to the coefficient of friction (μ) times the normal force (FN​).

  • Static Friction (v=0): Ff​=Fstiction​≤μstiction​FN​⋅sign(P), where P is the applied force.

  • Kinetic Friction (v=0): Ff​=Fkineticfriction​=μkineticfriction​FN​⋅sign(v).

• Viscous Friction:

  • Ff​=μvf​FN​v: Viscous friction force (Ff​) is proportional to the sliding velocity (v), normal force (FN​), and the viscous friction coefficient (μvf​).

03 Physics of Materials

• Friction Cone (Point on Plane Contact with Friction):

  • F=f∣∣∣ftangent​∣∣≤μs​∣∣fnormal​∣∣,fz​≥0: The set of admissible forces (F) where the magnitude of the tangential force (∣∣ftangent​∣∣) is limited by the static friction coefficient (μs​) and the magnitude of the normal force (∣∣fnormal​∣∣), and the normal force (fz​) must be non-negative.

  • Cone Angle: β=tan−1μs​.

• Soft-Finger Contact:

  • F=(f,τnormal​)∣∣∣ftangent​∣∣≤μs​∣∣fnormal​∣∣,fz​≥0,∣τnormal​∣≤γfz​: Similar to the friction cone, but also includes the transmission of normal torque (τnormal​) limited by a torsional friction coefficient (γ) and the normal force (fz​).

• Hooke’s Law (Spring):

  • −mg+kδ=0: In equilibrium, the force due to gravity (-mg) is balanced by the spring force (kδ), where k is the spring constant and δ is the displacement.

• Stress (σ):

  • σ=F/A0​: Normal stress is the applied force (F) divided by the original cross-sectional area (A0​).

• Strain (ϵ):

  • ϵ=Δl/l0​: Normal strain is the change in length (Δl) divided by the original length (l0​).

• Hooke’s Law (Material):

  • σ=Eϵ: Stress is linearly proportional to strain within the elastic region, where E is Young’s Modulus (the slope).

04 Joints

• Grübler’s Formula:

  • M=k(n−1)−∑i=1j​(k−fi​)=k(n−1−j)+∑i=1j​fi​: Calculates the degrees of freedom (M) of a mechanism, where k is the DOF of a link (3 for planar, 6 for spatial), n is the number of links (including ground), j is the number of joints, and fi​ is the degrees of freedom allowed by joint i.

• System Dynamics:

  • q¨​=f(q,q˙​,u,t): Represents the dynamics of a system, where q¨​ is the acceleration, q is the position, q˙​ is the velocity, u is the control input, and t is time. (Note: Slide 18 used q˙​ for acceleration, common notation uses q¨​).

05 Analog Electronics

• Current (i):

  • i(t)=dtdq(t)​≈ΔtΔq​: Current is the rate of change of charge (q) with respect to time (t).

• Ohm’s Law:

  • V=iR: Voltage (V) across a resistor equals the current (i) through it times its resistance (R).

• Resistance (R):

  • R=Aρl​: Resistance of a conductor depends on its resistivity (ρ), length (l), and cross-sectional area (A).

• Capacitor Current:

  • i=Cdtdv​: Current (i) through a capacitor is equal to its capacitance (C) times the rate of change of voltage (v) across it.

• Inductor Voltage:

  • vL​(t)=Ldtdi​: Voltage (vL​) across an inductor equals its inductance (L) times the rate of change of current (i) through it.

• Resistors in Series:

  • Req​=R1​+R2​+R3​+…: The equivalent resistance is the sum of individual resistances.

• Resistors in Parallel:

  • Req​=1/(1/R1​+1/R2​+1/R3​+…): The reciprocal of the equivalent resistance is the sum of the reciprocals of individual resistances.

• RC Circuit Discharge:

  • Vc​(t)=Vi​⋅e−t/RC: Voltage across a discharging capacitor (Vc​) decreases exponentially from an initial voltage (Vi​) with a time constant τ=RC.

• RC Circuit Charge:

  • Vc​(t)=Vs​(1−e−t/RC): Voltage across a charging capacitor (Vc​) increases exponentially towards the source voltage (Vs​) with a time constant τ=RC.

• RL Circuit (Charging/Current):

  • i(t)=RVs​​(1−e−Rt/L): Current through an inductor (i) increases exponentially towards its steady-state value (Vs​/R) with a time constant τ=L/R.

• RL Circuit (Charging/Voltage across L):

  • v(t)=Vs​e−Rt/L: Voltage across the inductor (v) decreases exponentially from the source voltage (Vs​) with a time constant τ=L/R.

• RLC Circuit Steady State (DC):

  • Capacitor acts as an open circuit (iC​=0).

  • Inductor acts as a short circuit (VL​=0).

• Differential Amplifier:

  • Vout​=A(V+−V−): Output voltage (Vout​) is the gain (A) multiplied by the difference between the non-inverting (V+) and inverting (V−) inputs.

• Power (P):

  • P=VI: Power is voltage (V) times current (I), measured in Watts.

06 Digital Electronics

• Boolean Algebra (NOT):

  • A=A: Double negation returns the original value.

• Boolean Algebra (AND):

  • A⋅A=A

  • A⋅1=A

  • A⋅0=0

  • A⋅A=0 (Contradiction)

  • A⋅B=B⋅A (Commutative)

  • A⋅(B⋅C)=(A⋅B)⋅C=ABC (Associative)

• Boolean Algebra (OR):

  • A+0=A

  • A+1=1

  • A+A=1 (Tautology)

  • A+A=A

  • (A+B)+C=A+(B+C)=A+B+C (Associative)

• Boolean Algebra (Distributive):

  • A⋅(B+C)=A⋅B+A⋅C

• Boolean Algebra Properties Summary: (Includes Laws like Commutative, Associative, Distributive, Absorption, etc. listed on slide 33)

• Example Combinatorial Logic Expression:

  • Q=(A⋅B)​⋅(A+B)​⋅C: Example Boolean expression for a specific logic circuit.

• Serial Adder Logic:

  • Y=ab+ay+by: Next state (carry-out) equation for a serial adder.

  • s=a⊕b⊕y: Output (sum) equation for a serial adder (XOR operation).

07 Actuators

• Torque-Speed Relationship (Linear Approximation):

  • τ=−k1​ω+k2​ or ω=−k1′​τ+k2′​: Describes the inverse linear relationship between motor torque (τ) and speed (ω).

• Power (Output):

  • P=τ⋅ω: Mechanical output power is torque times angular velocity.

• Efficiency (η):

  • η=Input PowerOutput Power​: Ratio of mechanical output power to electrical input power.

• Motor Circuit Model (Simplified):

  • Ldtdi​=−Ri−Ke​ω+v(t): Electrical equation relating voltage (v(t)), current (i), resistance (R), inductance (L), back EMF constant (Ke​), and speed (ω).

  • Iα=Kt​i−cω: Mechanical equation relating inertia (I), angular acceleration (α), torque constant (Kt​), current (i), damping (c), and speed (ω).

• Gear Train Mechanical Advantage (Simple Spur):

  • Mechanical Advantage=ω2​ω1​​=r1​r2​​=N1​N2​​=T1​T2​​: Ratio of output to input speed (ω), radius (r), number of teeth (N), or torque (T).

• Compound Gear Train Ratio:

  • Ndriven​Ndriver​​=Product of teeth on driver gearsProduct of teeth on driven gears​ (Generalized from examples like N6​N1​​=N1​N3​N5​N2​N4​N6​​ or N4​N1​​=N1​N3​N2​N4​​).

08 States and Behaviors

• Finite State Machine (FSM) Transition Function:

  • S×Σ→S: Defines how the current state (S) and an input from the alphabet (Σ) determine the next state (S).

• Markov Property:

  • p(st+1​∣st​,st−1​,st−2​,…)=p(st+1​∣st​): The probability of the next state (st+1​) depends only on the current state (st​).

09 Introduction to Control

• General State-Space Model (Continuous):

  • z˙=F(t,z,u): State derivative (z˙) is a function of time (t), state (z), and input (u).

  • y=G(t,z,u): Output (y) is a function of time (t), state (z), and input (u).

• Linear Time-Invariant (LTI) State-Space Model (Continuous):

  • dtdz​=Az+Bu: State derivative is a linear combination of state (z) and input (u).

  • y=Cz+Du: Output (y) is a linear combination of state (z) and input (u).

• General State-Space Model (Discrete):

  • zt+1​=F(zt​,ut​): Next state (zt+1​) is a function of the current state (zt​) and current input (ut​).

  • yt​=G(zt​): Current output (yt​) is a function of the current state (zt​).

• LTI State-Space Model (Discrete):

  • zt+1​=Azt​+But​: Next state is a linear combination of the current state and input.

  • yt​=Czt​: Current output is a linear combination of the current state.

• Control Error (e):

  • e=x−xd​ or e=xd​−x: The difference between the actual state/output (x) and the desired state/setpoint (xd​).

10 PID Control Strategies

• Cruise Control Model (Simplified):

  • mv˙=−bv+fhill​+fengine​: Equation of motion for car velocity (v) considering mass (m), damping (b), hill force (fhill​), and engine force (fengine​).

• Proportional Control Law (Simple):

  • fengine​=k(vdesired​−v): Engine force is proportional (gain k) to the velocity error.

• Steady State Velocity (Simple Proportional Control):

  • vSS​=b+kk​vdes​+b+k1​fhill​: Steady-state velocity depends on desired velocity, gain, damping, and disturbance.

• Engine Torque Model:

  • Te​=uTm​(1−β(ωm​ω​−1)2): Engine torque (Te​) depends on control input (u), max torque (Tm​), efficiency (β), current speed (ω), and speed at max torque (ωm​).

• Angular Speed to Velocity:

  • ω=rn​v=αn​v: Engine speed (ω) is related to car velocity (v) via gear ratio (n) and wheel radius (r), or combined gain (αn​).

• Force from Engine Torque:

  • F=αn​Te​(αn​,v): Force generated by wheels (F) is engine torque (Te​) multiplied by the gain (αn​).

• Disturbance Forces:

  • Fg​=mgsin(θ): Force due to gravity on a slope (θ).

  • Fr​=mgCr​sgn(v): Rolling friction force, dependent on coefficient (Cr​) and direction of velocity (sgn(v)).

  • Fa​=21​ρCd​Av2: Aerodynamic drag force, dependent on air density (ρ), drag coefficient (Cd​), frontal area (A), and velocity squared (v2).

• Full Cruise Control Dynamic Model (Nonlinear):

  • mdtdv​=αn​uTm​(1−β(ωm​αn​v​−1)2)−(mgsin(θ)+mgCr​sgn(v)+21​ρCd​Av2): Combines engine force and disturbance forces.

• Proportional (P) Control Law:

  • u(t)=kP​e(t): Control input is proportional to the current error. (Often includes saturation limits and bias/feedforward uff​).

• Proportional-Integral (PI) Control Law:

  • u(t)=kP​e(t)+kI​∫0t​e(τ)dτ: Adds an integral term to eliminate steady-state error.

• Proportional-Derivative (PD) Control Law:

  • u(t)=kP​e(t)+kD​dtde(t)​: Adds a derivative term to improve transient response and damping.

• Proportional-Integral-Derivative (PID) Control Law:

  • u(t)=kP​e(t)+kI​∫0t​e(τ)dτ+kD​dtde(t)​: Combines all three terms.

  • Alternative form: u(t)=K(e(t)+Ti​1​∫0t​e(τ)dτ+Td​dtde​) where K=Kp​, Ti​=Kp​/KI​, Td​=KD​/Kp​.

• Homogeneous 2nd Order ODE Solution:

  • q(t)=eαt(Acosωt+Bsinωt): General form of the solution for mq¨​+cq˙​+kq=0.

• Homogeneous 2nd Order Solution with ICs:

  • q(t)=e−ζωn​t[qo​cosωd​t+(ωd​ζωn​​qo​+ωd​1​vo​)sinωd​t]: Solution in terms of initial position (qo​), initial velocity (vo​), natural frequency (ωn​), damping ratio (ζ), and damped frequency (ωd​).

• Natural Frequency (ωn​):

  • ωn​=k/m​.

• Damping Ratio (ζ):

  • ζ=2km​c​=2mωn​c​.

• Damped Frequency (ωd​):

  • ωd​=ωn​1−ζ2​.

11 PID with State-Space Models

• 1-Compartment Drug Model:

  • Vdtdc​=−qc: Rate of change of concentration (c) depends on volume (V) and outflow rate (q).

  • dtdc​=−kc+bd​u: Simplified form with rate constants k and bd​, and input drug flow u.

• 2-Compartment Drug Model:

  • dtdc1​​=−(k0​+k1​)c1​+k2​c2​+b0​u: Dynamics for compartment 1 concentration (c1​).

  • dtdc2​​=k1​c1​−k2​c2​: Dynamics for compartment 2 concentration (c2​).

  • State-Space Form: dtdc​=[−(k0​+k1​)​k2​ k1​​−k2​​]c+[b0​ 0​]u.

• State Feedback Control Law:

  • u=−Kc​z+kr​yd​(t): Control input (u) is a linear combination of the negative state vector (z) scaled by gains (Kc​) and the desired output (yd​) scaled by a feedforward gain (kr​).

• Closed-Loop System Dynamics:

  • dtdz​=(A−BKc​)z+Bkr​yd​(t): The dynamics of the system under state feedback control.

• Characteristic Equation for Stability:

  • p(s)=det(sI−(A−BKc​))=0: The roots (eigenvalues λ) of this equation determine stability. For stability, the real part of all eigenvalues must be negative (Re(λi​)<0).

• 2nd Order System (Alternative Form):

  • x¨+2ζω0​x˙+ω02​x=0: Standard form using natural frequency (ω0​) and damping ratio (ζ).

12 Poses and Motion

• Point Representation:

  • p=[px​ py​ pz​​]∈R3: Represents a point p in 3D space using its coordinates.

• Vector 2-Norm (Magnitude):

  • ∣∣p∣∣2​=[p12​+p22​+…+pn2​]1/2: Calculates the Euclidean length (magnitude) of a vector p.

• Inner (Dot) Product:

  • ⟨p,q⟩=p⋅q=∑i=0n−1​pi​qi​=∣∣p∣∣2​∣∣q∣∣2​cos(θ): Computes the dot product of vectors p and q, related to the angle (θ) between them.

• Angle Between Vectors:

  • cos(θ)=∣∣p∣∣2​∣∣q∣∣2​p⋅q​: Calculates the cosine of the angle between vectors p and q using their dot product and magnitudes.

• Cross Product Magnitude:

  • ∣∣p×q∣∣=∣∣p∣∣∣∣q∣∣sin(θ): The magnitude of the cross product relates to the sine of the angle between vectors p and q.

• Cross Product (3D):

  • p×q=[p2​q3​−p3​q2​ p3​q1​−p1​q3​ p1​q2​−p2​q1​​]: Calculates the cross product vector for 3D vectors p and q.

• Matrix Exponential:

  • eA=I+A+2!A2​+3!A3​+…: Defines the matrix exponential using an infinite series.

• Translation:

  • p′=p+d: A translated point p′ is the original point p plus a displacement vector d.

• Rotation in 2D:

  • [x′ y′​]=[cosθ​−sinθ sinθ​cosθ​][x y​]: Rotates a 2D point (x, y) by angle θ.

• Basic 3D Rotation Matrices:

  • Rx​(θ)=[1​0​0 0​cosθ​−sinθ 0​sinθ​cosθ​]

  • Ry​(θ)=[cosθ​0​sinθ 0​1​0 −sinθ​0​cosθ​]

  • Rz​(θ)=[cosθ​−sinθ​0 sinθ​cosθ​0 0​0​1​]

• Properties of Rotation Matrices (R ∈ SO(3)):

  • R−1=RT

  • RTR=I

  • det(R)=1

• Rodrigues’ Rotation Formula:

  • x′=n^(n^⋅x)+sinθ(n^×x)−cosθ(n^×(n^×x))

• Skew-Symmetric Matrix (Cross Product Operator):

  • [n^]×​=[0​−nz​​ny​ nz​​0​−nx​ −ny​​nx​​0​]

• Rotation Matrix from Axis-Angle:

  • R=I+sinθ[n^]×+(1−cosθ)[n^]×2

• Quaternion Representation:

  • q=q0​+q1​i+q2​j+q3​k=(s,v)

• Quaternion Multiplication:

  • qq′=(ss′−v⋅v′,sv′+s′v+v×v′)

• Quaternion Inverse:

  • q−1=∣∣q∣∣22​(s,−v)​

• Rotation Matrix from Quaternion:

  • R=[1−2q22​−2q32​​2q1​q2​−2q0​q3​​2q1​q3​+2q0​q2​ 2q1​q2​+2q0​q3​​1−2q12​−2q32​​2q2​q3​−2q0​q1​ 2q1​q3​−2q0​q2​​2q2​q3​+2q0​q1​​1−2q12​−2q22​​]

• Quaternion from Rotation Matrix:

  • q0​=21​tr(M)+1​, q1​=4q0​m21​−m12​​, q2​=4q0​m02​−m20​​, q3​=4q0​m10​−m01​​

• Quaternion from Axis-Angle:

  • q=(cos2θ​,asin2θ​)

• Linear Interpolation (Lerp):

  • Lerp(t,a,b)=(1−t)a+tb

• Spherical Linear Interpolation (Slerp):

  • Slerp(t,a,b)=sinθsin((1−t)θ)​a+sinθsin(tθ)​b, where θ=cos−1(a⋅b).

13 Transformations

• General Displacement:

  • x′=Rx+d

• Homogeneous Transformation Matrix:

  • T=[R​p 0​1​]∈SE(3)

• Homogeneous Point:

  • p~​=[p 1​]

• Applying Transformation:

  • p​′=Tp​

• Transformation Matrix Inverse:

  • T−1=[RT​−RTp 0​1​]

• Composition of Transformations:

  • ATC​=ATB​BTC​

• Plücker Coordinates (Line Representation):

  • Line: x(t)=p+tq

  • Plücker coords: (q,q0​) where q0​=p×q

  • Constraint: q⋅q0​=0

• Spatial Velocity (Twist):

  • V=[ω v​]

• Adjoint Transformation (for Spatial Velocity):

  • AV=[ARB​​0 [ApBORG​]×​ARB​​ARB​​]BV

14 Forward and Inverse Kinematics

• Forward Kinematics (FK):

  • q∈Rn→x∈Rm

  • Example (2-Link Planar): x=l1​c1​+l2​c12​, y=l1​s1​+l2​s12​ (using c1​=cosθ1​, s12​=sin(θ1​+θ2​), etc.)

• Inverse Kinematics (IK):

  • x∈Rm→q∈Rn

• Jacobian Matrix (J):

  • x˙=J(q)q˙​

  • Definition: Jij​=∂qj​∂fi​​, where x=f(q).

• Velocity Inverse Kinematics:

  • q˙​=J−1x˙ (if J is square and invertible).

  • q˙​=J+x˙ (using pseudo-inverse).

• Condition Number:

  • κ(J)=σmin​σmax​​

• Jacobian Transpose (Statics):

  • τ=JTF

• Pseudo-Inverse:

  • J+=JT(JJT)−1 (for m<n)

  • J+=(JTJ)−1JT (for n<m)

• Denavit-Hartenberg (DH) Transformation:

  • i−1Ti​=Rot(zi−1​,θi​)Trans(zi−1​,di​)Trans(xi​,ai​)Rot(xi​,αi​)

  • Matrix form provided on slide 24.

16 Sensors and Vision

• Robot Equation of Motion (Lagrangian Formulation):

  • τ=M(q)q¨​+C(q,q˙​)q˙​+G(q)

• Pinhole Camera Model:

  • x′=fzx​, y′=fzy​

• Camera Projection with Offset:

  • (x,y,z)→(fzx​+cx​,fzy​+cy​)

• Homogeneous Coordinates: (See Slide 12)

• Camera Intrinsic Matrix (K - Homogeneous):

  • K=[fx​​0​cx​​0 0​fy​​cy​​0 0​0​1​0​]

• Radial and Tangential Distortion Model (Plumb Bob):

  • pc′​=pc​(1+k1​r2+k2​r4+k3​r6)+[2t1​xc​yc​+t2​(r2+2xc2​) t1​(r2+2yc2​)+2t2​xc​yc​​]

• Camera Projection (Full Model):

  • pc​=K⋅Tcb​⋅pb​

• Visual Servoing - Image Jacobian:

  • f˙​image​=Lξc​

17 Motion Planning

• A Search Heuristic:*

  • F=g+h

18 & 19 Machine Learning

• Linear Regression Model:

  • y=ax+b

• Sum of Squared Errors (SSE):

  • SSE=∑i=1n​(yi​−y^​i​)2

• Least Squares Solution (Matrix Form):

  • [∑xi2​​∑xi​ ∑xi​​n​][a b​]=[∑xi​yi​ ∑yi​​]

• Polynomial Regression Model:

  • y=an​xn+…+a1​x+a0​

• Least Squares Loss (Polynomial):

  • S=∑i=1N​(yi​−(an​xin​+…+a0​))2

• Bayes’ Rule:

  • P(Si​∣A)=∑j​P(Sj​)P(A∣Sj​)P(Si​)P(A∣Si​)​

• Naïve Bayes Classifier:

  • p(li​∣F1​,…,Fn​)∝p(F1​∣li​)⋅…⋅p(Fn​∣li​)⋅p(li​)

• Artificial Neuron Output:

  • o=f(∑i=1n​wi​xi​+b)

• Softmax Function:

  • y^​i​=∑j​eoj​eoi​​

• Markov Decision Process (MDP) Definition:

  • M=⟨S,A,P,R,γ⟩

• Transition Probability:

  • Pss′a=Pr[st+1=s′∣st​=s,at​=a]

• Reward Function:

  • r(s,a)=E[Rt+1​∣st​=s,at​=a]

• Policy (π):

  • π:S→A or π(a∣s)=Pr[At​=a∣St​=s]

• Goal (RL):

  • π∗=argmaxπ​E[∑t≥0​γtr(st​,π(st​))]

• Value Function (Vπ(s)):

  • Vπ(s)=E[∑k=0∞​γkRt+k+1​∣St​=s]

• Action-Value Function (Qπ(s,a)):

  • Qπ(s,a)=E[∑k=0∞​γkRt+k+1​∣St​=s,At​=a]

  • Qπ(s,a)=r(s,a)+γ∑s′∈S​P(s′∣s,a)Vπ(s′)

• Bellman Equation (for V):

  • Vπ(s)=∑a​π(a∣s)∑s′​∑r​p(s′,r∣s,a)[r+γVπ(s′)]

20 Introduction to HRI

(No specific mathematical equations presented in this slide deck).