Chapter 1: Linear Systems and Data Representations

I. Motivating Linear Systems for Roboticists

In autonomous systems, nearly every computation reduces to linear algebra operations:

• State representation: Pose of a robot, configuration of a manipulator arm
• Sensor models: Linear mappings from state to measurements (cameras, lidar)
• Dynamics linearization: Control design via feedback gains requires linearized models
• Estimation: Kalman filters, least-squares parameter fitting
• Coordinate transformations: Rotations, homogeneous transforms
• System identification: Learning dynamics models from data

This chapter develops the mathematical machinery—vectors, matrices, subspaces, eigenvalues— with robotics applications in mind. Unlike pure mathematics texts, we emphasize computational implementation and numerical stability from the start.

II. Data Representations

1. Vectors and Vector Spaces

Definition: Vector

A vector is an ordered list of numbers, denoted (\mathbf{v} \in \mathbb{R}^n): [\mathbf{v} = \begin{bmatrix} v_1 \ v_2 \ \vdots \ v_n \end{bmatrix}]

Geometrically: a point or direction in (n)-dimensional space. In robotics, vectors represent positions, velocities, forces, and sensor readings.

Vector Spaces and Axioms

A vector space (V) over a field (\mathbb{F}) (usually (\mathbb{R}) or (\mathbb{C})) is a set closed under two operations:

1. Vector Addition (\mathbf{u} + \mathbf{v}: V \times V \to V)

   • Commutative: (\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u})
   • Associative: ((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}))
   • Identity: (\exists , \mathbf{0} \in V : \mathbf{u} + \mathbf{0} = \mathbf{u})
   • Inverses: (\forall \mathbf{u} , \exists -\mathbf{u} : \mathbf{u} + (-\mathbf{u}) = \mathbf{0})

2. Scalar Multiplication (\alpha \mathbf{v}: \mathbb{F} \times V \to V)

   • Associativity: (\alpha(\beta \mathbf{v}) = (\alpha\beta) \mathbf{v})
   • Left distributive: (\alpha(\mathbf{u} + \mathbf{v}) = \alpha\mathbf{u} + \alpha\mathbf{v})
   • Right distributive: ((\alpha + \beta)\mathbf{v} = \alpha\mathbf{v} + \beta\mathbf{v})
   • Unit scaling: (1 \cdot \mathbf{v} = \mathbf{v})

Key Examples:

• (\mathbb{R}^n): (n)-dimensional real Euclidean space (most robotics applications)
• (\mathbb{C}^n): complex vectors (frequency-domain analysis, eigenvalues)
• Function spaces: (\mathcal{C}[0,T]) = continuous functions on ([0,T]) (state trajectories)

Linear Functions (Functionals)

Theorem (Riesz Representation): If (f: \mathbb{R}^n \to \mathbb{R}) is linear, then [f(\mathbf{x}) = \mathbf{a}^T \mathbf{x}] for some unique (\mathbf{a} \in \mathbb{R}^n).

Proof Sketch: For the standard basis (\mathbf{e}i = (0, \ldots, 1, \ldots, 0)^T), [f(\mathbf{x}) = f\left(\sum{i=1}^n x_i \mathbf{e}i\right) = \sum{i=1}^n x_i f(\mathbf{e}_i) = \mathbf{a}^T \mathbf{x}] where (a_i := f(\mathbf{e}_i)).

Superposition Principle: Linear functions satisfy [f(\alpha \mathbf{u} + \beta \mathbf{v}) = \alpha f(\mathbf{u}) + \beta f(\mathbf{v})]

Affine Functions: A function is affine if it is linear plus a constant: [f(\mathbf{x}) = \mathbf{a}^T \mathbf{x} + b]

Equivalently: (f(\alpha \mathbf{u} + \beta \mathbf{v}) = \alpha f(\mathbf{u}) + \beta f(\mathbf{v})) whenever (\alpha + \beta = 1).

Application: Robot dynamics after feedback (u = -K x + r) is affine in state.

Norms: Measuring Vector "Size"

A norm (|\cdot|: V \to \mathbb{R}_{\geq 0}) satisfies:

1. Positive Definiteness: (|\mathbf{v}| > 0) unless (\mathbf{v} = \mathbf{0})
2. Homogeneity: (|\alpha \mathbf{v}| = |\alpha| |\mathbf{v}|)
3. Triangle Inequality: (|\mathbf{u} + \mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}|)

(L_p) Norms

For (p \geq 1), the (L_p) norm is: [|\mathbf{v}|p = \left(\sum{i=1}^{n} |v_i|^p\right)^{1/p}]

Euclidean Norm (Most Common)

[|\mathbf{v}|_2 = \sqrt{\mathbf{v}^T \mathbf{v}} = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}]

Geometric interpretation: Euclidean distance from origin to point (\mathbf{v}).

Inner Product Definition: [\mathbf{u}^T \mathbf{v} = |\mathbf{u}|_2 |\mathbf{v}|_2 \cos\theta] where (\theta) is the angle between vectors. Orthogonal vectors satisfy (\mathbf{u}^T \mathbf{v} = 0).

2. Matrices and Linear Transformations

Definition: Matrix

An (m \times n) matrix is a rectangular array: [A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}]

• Rows: (m) (vertical dimension)
• Columns: (n) (horizontal dimension)
• Entry: (A_{ij}) or (a_{ij}) is the element at row (i), column (j)

Special Cases:

• Column vector: (n \times 1) matrix
• Row vector: (1 \times n) matrix
• Square matrix: (m = n)
• Tall matrix: (m > n) (more rows than columns; often overdetermined systems)
• Wide matrix: (m < n) (more columns than rows; underdetermined systems)

Application: In sensor fusion, an (m \times n) Jacobian matrix relates joint velocities ((n) DOF) to end-effector velocities ((m) task dimensions).

Linear Transformations

A linear transformation (T: \mathbb{R}^n \to \mathbb{R}^m) satisfies:

1. (T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v}))
2. (T(\alpha \mathbf{v}) = \alpha T(\mathbf{v}))

Key Theorem: Every linear transformation (T: \mathbb{R}^n \to \mathbb{R}^m) can be represented by a unique (m \times n) matrix (A) such that [T(\mathbf{x}) = A\mathbf{x}]

The columns of (A) are the images of the standard basis: [A = \begin{bmatrix} | & & | \ T(\mathbf{e}_1) & \cdots & T(\mathbf{e}_n) \ | & & | \end{bmatrix}]

Matrix Operations

Addition (only for same size): [(A + B){ij} = A{ij} + B_{ij}]

Scalar Multiplication: [(cA){ij} = c \cdot A{ij}]

Matrix Multiplication (compatible dimensions: (A) is (m \times n), (B) is (n \times p)): [(AB){ij} = \sum{k=1}^{n} A_{ik} B_{kj}]

Critical Note: Matrix multiplication is not commutative: (AB \neq BA) in general.

Matrix Inverse (for square (A), if it exists): [A^{-1} : A A^{-1} = A^{-1} A = I_n]

Exists if and only if (\text{rank}(A) = n) (full rank).

Properties:

• ((AB)^{-1} = B^{-1} A^{-1})
• ((A^T)^{-1} = (A^{-1})^T)

Transpose

[(A^T){ij} = A{ji}]

Properties:

• ((A^T)^T = A)
• ((A + B)^T = A^T + B^T)
• ((AB)^T = B^T A^T)
• ((A^{-1})^T = (A^T)^{-1})

Application: In control, the dual system (\dot{x} = A^T x) characterizes observability.

Span, Basis, and Dimension

Linear Combination: (\mathbf{v}) is a linear combination of ({\mathbf{v}_1, \ldots, \mathbf{v}_k}) if [\mathbf{v} = c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k] for some scalars (c_i).

Span: The set of all linear combinations: [\text{span}{\mathbf{v}_1, \ldots, \mathbf{v}k} = \left{ \sum{i=1}^k c_i \mathbf{v}_i : c_i \in \mathbb{R} \right}]

Linear Independence: Vectors ({\mathbf{v}_1, \ldots, \mathbf{v}_k}) are linearly independent if [c_1 \mathbf{v}_1 + \cdots + c_k \mathbf{v}_k = \mathbf{0} \quad \Rightarrow \quad c_i = 0 \text{ for all } i]

Otherwise, they are linearly dependent.

Basis: A set (B) is a basis for vector space (V) if:

1. (B) is linearly independent
2. (\text{span}(B) = V)

Dimension: The number of vectors in any basis for (V): [\dim(V) = |B|]

Application: In a robot with 6 DOF in (\text{SE}(3)), any valid configuration space basis has dimension 6.

Rank and Nullity

For a matrix (A \in \mathbb{R}^{m \times n}):

• Rank (\text{rank}(A) = r) = dimension of the column space
• Nullity (\text{nullity}(A) = n - r) = dimension of the null space

Rank-Nullity Theorem: [\text{rank}(A) + \text{nullity}(A) = n]

Application: For a robot Jacobian (J \in \mathbb{R}^{6 \times n}):

• (\text{rank}(J) = 6) everywhere except singularities (full rank = fully controllable end-effector)
• (\text{nullity}(J) > 0) at singularities (null space = redundant DOF)

Determinants

The determinant (\det(A)) of a square (n \times n) matrix encodes crucial information: invertibility, volume scaling, and eigenvalues.

Leibniz Formula: [\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^{n} A_{i, \sigma(i)}]

where (S_n) is all permutations of ({1, \ldots, n}) and (\text{sgn}(\sigma) = \pm 1).

Cofactor Expansion (easier to compute): [\det(A) = \sum_{j=1}^{n} (-1)^{1+j} A_{1j} \det(A_{1j})] where (A_{1j}) is the ((n-1) \times (n-1)) submatrix obtained by deleting row 1 and column (j).

Key Properties:

1. (\det(AB) = \det(A)\det(B))
2. (\det(A^T) = \det(A))
3. (\det(cA) = c^n \det(A))
4. (\det(A) = 0 \iff A) is singular (non-invertible)
5. (\det(A) = \prod_{i=1}^{n} \lambda_i) where (\lambda_i) are eigenvalues

III. Solving Linear Systems and Subspaces

The Four Fundamental Subspaces

For any (A \in \mathbb{R}^{m \times n}) with rank (r):

Orthogonality Relations:

• (C(A^T) \perp N(A)): row space orthogonal to null space
• (C(A) \perp N(A^T)): column space orthogonal to left null space

Direct Sum Decomposition:

• (\mathbb{R}^n = C(A^T) \oplus N(A)) (every vector = controllable part + uncontrollable part)
• (\mathbb{R}^m = C(A) \oplus N(A^T))

Application: In observability analysis, the left null space of the observability matrix characterizes unobservable modes.

Solving (A\mathbf{x} = \mathbf{b})

Three cases:

1. Square, full rank ((m = n), (\det(A) \neq 0)): Unique solution (\mathbf{x} = A^{-1}\mathbf{b})
2. Overdetermined ((m > n)): Usually no exact solution; find least-squares (\mathbf{\hat{x}})
3. Underdetermined ((m < n)): Infinitely many solutions; find minimum-norm solution

Gaussian Elimination (classical approach):

1. Augment: ([A | \mathbf{b}])
2. Forward elimination to row echelon form (REF)
3. Back substitution

Numerically Robust: Use (A = QR) decomposition (QR method) or (A = U\Sigma V^T) (SVD).

IV. Eigenvalues, Eigenvectors, and Spectral Analysis

Eigenvalue Equation

For a square matrix (A), the equation [A\mathbf{v} = \lambda \mathbf{v}] has a non-trivial solution if and only if (\det(A - \lambda I) = 0).

• Eigenvalue (\lambda): scalar
• Eigenvector (\mathbf{v} \neq \mathbf{0}): vector

Geometric Meaning: (A) only scales the eigenvector by factor (\lambda); the direction is unchanged.

Characteristic Polynomial

[p(\lambda) = \det(A - \lambda I) = (-1)^n \lambda^n + (\text{lower order terms})]

• Degree: (n) (an (n \times n) matrix has (n) eigenvalues, counting multiplicities and complex values)
• Roots: eigenvalues (\lambda_1, \ldots, \lambda_n)

Application: Stability of continuous-time LTI system (\dot{\mathbf{x}} = A\mathbf{x}) is determined by Re((\lambda_i) < 0) for all (i).

Diagonalization

If (A) has (n) linearly independent eigenvectors (collected as columns of (S)), then [A = S\Lambda S^{-1}] where (\Lambda = \text{diag}(\lambda_1, \ldots, \lambda_n)).

When is this possible?

• All eigenvalues are distinct: always diagonalizable
• Repeated eigenvalues: diagonalizable if geometric multiplicity = algebraic multiplicity (i.e., enough independent eigenvectors)
• Defective matrices: Cannot be diagonalized; use Jordan normal form instead

Power of (A): [A^k = S\Lambda^k S^{-1}]

Application: Discrete-time system (\mathbf{x}_{k+1} = A\mathbf{x}_k) solutions: (\mathbf{x}_k = A^k \mathbf{x}_0 = S\Lambda^k S^{-1} \mathbf{x}_0).

Spectral Theorem (Symmetric Matrices)

If (A = A^T) (symmetric), then:

1. All eigenvalues are real
2. Eigenvectors form an orthonormal basis
3. (A = Q\Lambda Q^T) where (Q) is orthogonal ((Q^T Q = I))

Consequence: Symmetric matrices are always diagonalizable.

Application: Covariance matrices in Kalman filters are symmetric positive semidefinite; eigendecomposition is numerically stable.

Spectral Radius

[\rho(A) = \max_i |\lambda_i|]

• Stability: Continuous-time (\dot{\mathbf{x}} = A\mathbf{x}) is stable if (\rho(A) < 0) (all Re((\lambda_i) < 0))
• Discrete-time: (\mathbf{x}_{k+1} = A\mathbf{x}_k) is stable if (\rho(A) < 1)

V. Orthogonality and Least-Squares

Orthogonal Vectors and Matrices

Orthogonality: (\mathbf{u} \perp \mathbf{v}) if (\mathbf{u}^T \mathbf{v} = 0).

Orthogonal Set: Vectors ({\mathbf{v}_1, \ldots, \mathbf{v}_k}) are mutually orthogonal if (\mathbf{v}_i^T \mathbf{v}_j = 0) for (i \neq j).

Orthonormal Set: Orthogonal + unit norm: (|\mathbf{v}_i|_2 = 1) and (\mathbf{v}_i^T \mathbf{v}j = \delta{ij}).

Orthogonal Matrices

A square matrix (Q \in \mathbb{R}^{n \times n}) is orthogonal if its columns are orthonormal: [Q^T Q = Q Q^T = I_n]

Equivalently: (Q^T = Q^{-1}).

Properties:

• (|Q\mathbf{x}|_2 = |\mathbf{x}|_2) (preserves Euclidean norm)
• (\det(Q) = \pm 1)
• Numerically stable: no amplification of rounding errors

Application: Rotation matrices are orthogonal. Coordinate transformations via (\mathbf{x}{\text{new}} = R\mathbf{x}{\text{old}}) preserve distances.

Gram–Schmidt Orthogonalization

Given linearly independent vectors ({\mathbf{a}_1, \ldots, \mathbf{a}_n}), produce orthonormal vectors ({\mathbf{q}_1, \ldots, \mathbf{q}_n}):

Algorithm:

for j = 1 to n:
    v_j = a_j
    for i = 1 to j-1:
        v_j = v_j - (q_i^T a_j) * q_i  // subtract projection onto q_i
    q_j = v_j / ||v_j||_2             // normalize

Foundation of QR decomposition: (A = QR) where (Q) has orthonormal columns and (R) is upper triangular.

Least-Squares and Projection

Problem: Solve overdetermined system (A\mathbf{x} = \mathbf{b}) where (m > n) and (\mathbf{b} \notin C(A)).

Least-Squares Solution: Find (\mathbf{\hat{x}}) minimizing [|\mathbf{e}|_2^2 = |A\mathbf{x} - \mathbf{b}|_2^2]

Normal Equations: [A^T A \mathbf{\hat{x}} = A^T \mathbf{b}]

(Solvable if (A) has full column rank.)

Projection Matrix (for full column rank): [P = A(A^T A)^{-1}A^T]

Projects any vector onto (C(A)):

• (\mathbf{p} = P\mathbf{b}) = best approximation to (\mathbf{b}) in (C(A))
• (\mathbf{e} = \mathbf{b} - \mathbf{p} = P^{\perp}\mathbf{b}) where (P^{\perp} = I - P)

Residual Vector: (\mathbf{e} = \mathbf{b} - A\mathbf{\hat{x}}) satisfies (A^T \mathbf{e} = \mathbf{0}) (residual is orthogonal to column space).

Application: Robot sensor calibration via least-squares parameter estimation from noisy measurements.

Pseudo-Inverse

For (A \in \mathbb{R}^{m \times n}) with full column rank: [A^{\dagger} = (A^T A)^{-1}A^T]

The least-squares solution is (\mathbf{\hat{x}} = A^{\dagger} \mathbf{b}).

General pseudo-inverse (via SVD, works for all ranks): [A^{\dagger} = V \Sigma^{\dagger} U^T] where (A = U\Sigma V^T) and (\Sigma^{\dagger}) inverts non-zero singular values.

VI. Matrix Factorizations

QR Decomposition

Form: (A = QR)

• (Q): (m \times n) matrix with orthonormal columns
• (R): (n \times n) upper triangular matrix

When to use:

• Numerically stable least-squares: solve (R\mathbf{x} = Q^T\mathbf{b}) by back-substitution
• Gram–Schmidt orthogonalization

Computational cost: (\sim 2mn^2) flops (Householder reflections or Givens rotations).

Singular Value Decomposition (SVD)

Form: (A = U\Sigma V^T)

• (U): (m \times m) orthogonal matrix
• (\Sigma): (m \times n) diagonal matrix with non-negative singular values (\sigma_1 \geq \sigma_2 \geq \cdots \geq \sigma_{\min(m,n)} \geq 0)
• (V): (n \times n) orthogonal matrix

Key properties:

• (\text{rank}(A) = #) of non-zero singular values
• (\text{cond}(A) = \sigma_1 / \sigma_n) (condition number)
• Null space: columns of (V) corresponding to zero singular values
• Column space: columns of (U) corresponding to non-zero singular values

Applications:

• Computing pseudo-inverse: (A^{\dagger} = V\Sigma^{\dagger}U^T)
• Condition number assessment (numerical stability)
• Rank detection and truncation (data compression)
• Principal Component Analysis (PCA)

Application: In feature-based SLAM, SVD of measurement-to-state Jacobian detects observable directions.

LU Decomposition (with Partial Pivoting)

Form: (PA = LU)

• (P): permutation matrix (row swaps for numerical stability)
• (L): (n \times n) lower triangular with 1's on diagonal
• (U): (n \times n) upper triangular

When to use:

• Solving multiple systems with same (A): (PA = LU) once, then solve (L(U\mathbf{x}) = P\mathbf{b}) for each (\mathbf{b})
• Efficient for sparse matrices

Computational cost: (\sim n^3/3) flops.

Cholesky Decomposition

Form: (A = LL^T)

• (L): lower triangular with positive diagonal entries
• (A) must be symmetric positive definite (SPD)

When to use:

• Fastest factorization for SPD matrices (half the flops of LU)
• Solving (A\mathbf{x} = \mathbf{b}) with SPD (A): forward/back-substitution

Computational cost: (\sim n^3/6) flops.

Application: Covariance matrix factorization in Kalman filters. Cholesky is numerically stable and efficient.

Numerical Note: If Cholesky fails (negative diagonal pivot), matrix is not SPD—check for rounding errors or ill-conditioning.

Schur and Jordan Normal Forms

Schur Decomposition (for any square matrix): [A = QTQ^T] where (Q) is orthogonal and (T) is upper triangular (real Schur form uses block-triangular).

Jordan Normal Form (non-symmetric): [A = SJS^{-1}] where (J) is block-diagonal with Jordan blocks. Used to handle repeated eigenvalues and defective matrices.

VII. Jacobians and Taylor Approximations

Jacobian Matrix

For a function (\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m), the Jacobian is: [J_{\mathbf{f}}(\mathbf{x}) = \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \ \vdots & \ddots & \vdots \ \frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \end{bmatrix}]

Size: (m \times n)

Meaning: (J_{\mathbf{f}}(\mathbf{x}_0)) is the best linear approximation of (\mathbf{f}) near (\mathbf{x}_0).

Application: Robot Jacobian (J(\mathbf{q})) relates joint velocities to end-effector velocities: [\mathbf{v}_{\text{EE}} = J(\mathbf{q}) \dot{\mathbf{q}}]

First-Order Taylor Approximation

For (\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m) differentiable near (\mathbf{x}_0): [\hat{\mathbf{f}}(\mathbf{x}) = \mathbf{f}(\mathbf{x}0) + J{\mathbf{f}}(\mathbf{x}_0)(\mathbf{x} - \mathbf{x}_0)]

Accuracy: Excellent when all (x_i - x_{0i}) are small.

Application: Linearization for control design. For nonlinear dynamics (\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u})), linearize around equilibrium to design LTI feedback.

VIII. Practical Example: Robot Sensor Calibration via Least-Squares

Problem: A 6-axis force/torque sensor has unknown biases (\mathbf{b}) and a scale matrix (M). Collect (N) measurements under known loads and estimate ([\mathbf{b}, M]).

Model: [\mathbf{y}_i = M \mathbf{f}_i + \mathbf{b} + \boldsymbol{\epsilon}_i]

where (\mathbf{y}_i) = measured force/torque, (\mathbf{f}_i) = true load, (\boldsymbol{\epsilon}_i \sim \mathcal{N}(0, \Sigma)) noise.

Rewritten as linear system: Stack vertically to get (A\mathbf{x} = \mathbf{y} + \boldsymbol{\epsilon}) where (\mathbf{x}) contains the unknown parameters (36 scale elements + 6 biases = 42 parameters).

Solution:

# Collect N measurements (typically N > 50 for robustness)
A = [ones(N, 1)  f_data]  # Design matrix: [1 f_x f_y f_z ...]
y = measurements         # Stacked sensor readings

# Normal equations
x_hat = (A' * A) \ (A' * y)

# Extract bias and scale
bias_hat = x_hat[1:6]
scale_hat = reshape(x_hat[7:end], 6, 6)

# Check residuals
residuals = y - A * x_hat
rmse = sqrt(mean(residuals.^2))

Numerical Considerations:

• If (\text{cond}(A^T A)) is large, use SVD: (\text{x_hat} = (V \Sigma^{\dagger} U^T) \mathbf{y})
• Add regularization if underdetermined: ((A^T A + \lambda I)\mathbf{x} = A^T \mathbf{y})
• Validate on held-out test loads

Summary

Linear algebra is the mathematical foundation of autonomous systems. Key takeaways:

Next Steps: These tools are used throughout autonomous systems—in dynamics (rigid-body models), estimation (Kalman filters), and control (feedback design). The following chapters build on this foundation.

Chapter 2: Probability and Information Theory

I. The Role of Uncertainty in Robotics

Real robots operate in noisy, uncertain environments. Unlike laboratory simulations, autonomous systems must:

1. Sense with imperfect sensors (GPS drift, camera noise, lidar artifacts)
2. Act with noisy actuators (motor slip, compliance, delays)
3. Reason about incomplete information (partial observations, dynamic environments)

Probabilistic Robotics replaces point estimates with probability distributions, allowing systems to:

• Quantify uncertainty explicitly
• Make principled decisions under risk
• Fuse information from multiple sensors
• Adapt as uncertainty decreases

Sources of Uncertainty

II. Foundations of Probability

Probability Spaces and Axioms

A probability space ((\Omega, \mathcal{F}, P)) consists of:

• Sample space (\Omega): set of all possible outcomes
• (\sigma)-algebra (\mathcal{F}): collection of events (measurable subsets), closed under countable unions and complements
• Probability measure (P: \mathcal{F} \to [0, 1]): assigns probability to events

Axioms:

1. (P(\Omega) = 1) (certainty)
2. (P(\emptyset) = 0) (impossibility)
3. Countable additivity: if events are disjoint, (P(\cup_i A_i) = \sum_i P(A_i))

Random Variables

A random variable (X: \Omega \to \mathbb{R}) is a function assigning real values to outcomes.

Cumulative Distribution Function (CDF): [F_X(x) = P(X \leq x)]

Probability Mass Function (PMF) (discrete): [p_X(x) = P(X = x)]

Probability Density Function (PDF) (continuous): [f_X(x) = \frac{dF_X}{dx}]

with (\int_{-\infty}^{\infty} f_X(x) dx = 1).

Conditional Probability and Bayes' Rule

Conditional Probability: [P(A | B) = \frac{P(A \cap B)}{P(B)}]

assuming (P(B) > 0).

Bayes' Rule (foundation of probabilistic inference): [P(\theta | z) = \frac{P(z | \theta) P(\theta)}{P(z)}]

where:

• (P(\theta | z)): posterior (updated belief after observing (z))
• (P(z | \theta)): likelihood (how well (\theta) explains the data)
• (P(\theta)): prior (pre-data belief)
• (P(z) = \int P(z | \theta) P(\theta) d\theta): evidence (normalizing constant)

Application: In robot localization, Bayes' rule updates the belief over robot pose given range measurements.

Independence and Conditional Independence

Independence: Events (A) and (B) are independent if (P(A \cap B) = P(A)P(B)), written (A \perp B).

Conditional Independence: (A \perp B | C) if (P(A \cap B | C) = P(A | C)P(B | C)).

Factorization: Conditional independence enables decomposing complex joint distributions: [P(X, Y, Z) = P(X | Y, Z) P(Y | Z) P(Z)]

Application: Markov assumption in particle filters: current state conditionally independent of history given previous state.

III. Expectation, Variance, and Covariance

Expectation

The expectation (mean, expected value) is:

• Discrete: (\mathbb{E}[X] = \sum_x x , p_X(x))
• Continuous: (\mathbb{E}[X] = \int_{-\infty}^{\infty} x f_X(x) dx)

Linearity of Expectation (even for dependent variables): [\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]]

Variance

Variance measures spread: [\text{Var}(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] = \mathbb{E}[X^2] - (\mathbb{E}[X])^2]

Properties:

• (\text{Var}(aX + b) = a^2 \text{Var}(X))
• For independent: (\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y))

Covariance and Correlation

Covariance: [\text{Cov}(X, Y) = \mathbb{E}[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])]]

Correlation: [\rho(X, Y) = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X)\text{Var}(Y)}} \in [-1, 1]]

Covariance Matrix (for random vector (\mathbf{X} \in \mathbb{R}^n)): [\Sigma = \mathbb{E}[(\mathbf{X} - \boldsymbol{\mu})(\mathbf{X} - \boldsymbol{\mu})^T]]

where (\boldsymbol{\mu} = \mathbb{E}[\mathbf{X}]). Properties:

• Symmetric: (\Sigma^T = \Sigma)
• Positive semidefinite: (\Sigma \succeq 0)

Application: In Kalman filters, the covariance matrix (\Sigma) quantifies estimation uncertainty; its eigenvalues reveal principal axes of uncertainty.

IV. Key Probability Distributions

Gaussian (Normal) Distribution

PDF: [f_X(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)]

denoted (\mathcal{N}(\mu, \sigma^2)).

Properties:

• Symmetric around (\mu)
• (68.3%) of mass within (1\sigma), (95.4%) within (2\sigma), (99.7%) within (3\sigma) (68–95–99.7 rule)
• Preserved under linear transformations: if (X \sim \mathcal{N}(\mu, \Sigma)), then (AX + b \sim \mathcal{N}(A\mu + b, A\Sigma A^T))

Multivariate Gaussian ((\mathbf{X} \in \mathbb{R}^n)): [f(\mathbf{x}) = \frac{1}{\sqrt{(2\pi)^n \det(\Sigma)}} \exp\left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^T \Sigma^{-1} (\mathbf{x}-\boldsymbol{\mu})\right)]

denoted (\mathcal{N}(\boldsymbol{\mu}, \Sigma)).

Why Gaussian?

• Central Limit Theorem: Sum of many independent random variables (\approx) Gaussian
• Tractable: Closed-form Bayesian inference, linear transformations
• Sensor noise: Often well-modeled as Gaussian
• Computational: Maximum entropy distribution with specified mean and covariance

Application: Kalman filter assumes Gaussian distributions; nonlinear systems use Extended Kalman Filter (EKF) with local Gaussian approximations.

Other Important Distributions

V. Fundamental Theorems

Law of Large Numbers (LLN)

Weak LLN: For i.i.d. samples (X_1, X_2, \ldots) with mean (\mu), [\bar{X}n = \frac{1}{n}\sum{i=1}^n X_i \xrightarrow{p} \mu \quad \text{as } n \to \infty]

(convergence in probability)

Justification for Monte Carlo: empirical average converges to true expectation.

Central Limit Theorem (CLT)

For i.i.d. samples (X_i) with mean (\mu) and variance (\sigma^2), [\frac{\bar{X}_n - \mu}{\sigma / \sqrt{n}} \xrightarrow{d} \mathcal{N}(0, 1)]

(convergence in distribution to standard normal)

Consequence: Confidence interval for (\mu): [\bar{X}n \pm z{\alpha/2} \frac{\sigma}{\sqrt{n}}]

where (z_{\alpha/2}) is the critical value (e.g., (1.96) for (95%)).

Why CLT matters in robotics: Aggregate measurement noise (from many sources) is often approximately Gaussian even if individual sources are not.

Markov and Chebyshev Inequalities

Markov Inequality (for (X \geq 0)): [P(X \geq a) \leq \frac{\mathbb{E}[X]}{a}]

Chebyshev Inequality: [P(|X - \mu| \geq k\sigma) \leq \frac{1}{k^2}]

Use: Model-free bounds when distribution is unknown. Conservative but guaranteed.

VI. Bayesian Inference and Estimation

Maximum Likelihood Estimation (MLE)

Given observations (\mathbf{z} = {z_1, \ldots, z_n}) and model with unknown parameters (\boldsymbol{\theta}), the MLE is: [\hat{\boldsymbol{\theta}}{\text{MLE}} = \arg\max{\boldsymbol{\theta}} P(\mathbf{z} | \boldsymbol{\theta})]

or equivalently (log-likelihood): [\hat{\boldsymbol{\theta}}{\text{MLE}} = \arg\max{\boldsymbol{\theta}} \sum_{i=1}^n \log P(z_i | \boldsymbol{\theta})]

Properties:

• Asymptotically unbiased and efficient
• No prior information needed
• Often has closed-form solution (e.g., Gaussian likelihood (\Rightarrow) least-squares)

Application: Sensor calibration (estimate bias, scale).

Maximum A Posteriori (MAP) Estimation

Incorporates prior belief: [\hat{\boldsymbol{\theta}}{\text{MAP}} = \arg\max{\boldsymbol{\theta}} P(\boldsymbol{\theta} | \mathbf{z}) = \arg\max_{\boldsymbol{\theta}} P(\mathbf{z} | \boldsymbol{\theta}) P(\boldsymbol{\theta})]

Relationship to MLE: (\text{MAP} = \text{MLE}) when prior is uniform.

Regularization: Informative prior acts as regularization (penalizes unlikely parameters).

Conjugate Priors

If prior (P(\boldsymbol{\theta})) and likelihood (P(\mathbf{z} | \boldsymbol{\theta})) are conjugate, the posterior has the same functional form as the prior. Enables sequential Bayesian updates.

Example: Beta-Binomial conjugacy

• Prior: (P(p) = \text{Beta}(\alpha, \beta))
• Likelihood: (P(z | p) = \text{Binomial}(n, p))
• Posterior: (P(p | z) = \text{Beta}(\alpha + \text{successes}, \beta + \text{failures}))

Application: Sequential robot learning with minimal re-computation.

VII. Information Theory

Entropy

Entropy quantifies average information (or uncertainty): [H[X] = -\sum_x P(X=x) \log P(X=x)]

(continuous: (H[X] = -\int f(x) \log f(x) dx))

Interpretation:

• High entropy: uniform distribution, high uncertainty
• Low entropy: concentrated distribution, low uncertainty
• Minimum: (H = 0) for deterministic variable (one outcome has probability 1)

Units: bits (log base 2) or nats (natural log).

Mutual Information

Mutual Information measures dependence: [I[X; Y] = H[X] + H[Y] - H[X, Y]]

or equivalently: [I[X; Y] = \mathbb{E}_{X,Y}\left[\log \frac{P(X,Y)}{P(X)P(Y)}\right]]

Interpretation:

• (I[X; Y] = 0): (X) and (Y) are independent
• (I[X; Y] > 0): learning (Y) reduces uncertainty about (X)

Application: Information gain in active sensing; choose measurements that maximize information about unobserved state.

Kullback–Leibler Divergence

KL divergence measures distribution distance: [D_{\text{KL}}(P | Q) = \sum_x P(x) \log \frac{P(x)}{Q(x)} = \mathbb{E}_P[\log P(X) - \log Q(X)]]

Properties:

• Non-symmetric: (D_{\text{KL}}(P | Q) \neq D_{\text{KL}}(Q | P))
• Non-negative: (D_{\text{KL}}(P | Q) \geq 0), with equality iff (P = Q)
• Not a true distance (fails triangle inequality)

Application: Variational inference; approximate intractable posterior (P(\boldsymbol{\theta} | \mathbf{z})) with tractable (Q(\boldsymbol{\theta})) by minimizing (D_{\text{KL}}(Q | P)).

VIII. Practical Example: Estimating Sensor Noise from Stationary Data

Problem: Characterize noise statistics of a gyroscope when robot is stationary.

Setup: Collect (n = 1000) samples over time (T = 100) s. True angular velocity (\omega_{\text{true}} = 0) (stationary).

Assume Gaussian noise: (z_i \sim \mathcal{N}(0, \sigma^2)).

Estimators: [\hat{\mu} = \frac{1}{n}\sum_i z_i, \quad \hat{\sigma}^2 = \frac{1}{n-1}\sum_i (z_i - \hat{\mu})^2]

Confidence interval on variance (using chi-squared distribution): [\frac{(n-1)\hat{\sigma}^2}{\chi^2_{1-\alpha/2, n-1}} \leq \sigma^2 \leq \frac{(n-1)\hat{\sigma}^2}{\chi^2_{\alpha/2, n-1}}]

Implementation (Python):

import numpy as np
from scipy.stats import chi2

# Simulate gyro data
true_variance = 0.01  # rad/s)^2
gyro_data = np.random.normal(0, np.sqrt(true_variance), n=1000)

# Estimate mean and variance
mu_hat = np.mean(gyro_data)
sigma_hat = np.std(gyro_data, ddof=1)

# 95% confidence interval
n = len(gyro_data)
chi2_lower = chi2.ppf(0.025, n-1)
chi2_upper = chi2.ppf(0.975, n-1)

var_ci_lower = (n-1) * sigma_hat**2 / chi2_upper
var_ci_upper = (n-1) * sigma_hat**2 / chi2_lower

print(f"Estimated variance: {sigma_hat**2:.6f}")
print(f"95% CI: [{var_ci_lower:.6f}, {var_ci_upper:.6f}]")

# Check if true value is in CI
if var_ci_lower <= true_variance <= var_ci_upper:
    print("✓ True variance is in confidence interval")
else:
    print("✗ True variance outside CI (check for systematic bias)")

Summary

Probability and information theory are essential for handling uncertainty in autonomous systems:

Next Chapter: We apply these probability foundations to state estimation—the core task of perception in autonomous systems.