Overview of Robotics and Key Topics

Historical Overview

Robotics is an interdisciplinary field spanning mechanical engineering, electrical engineering, computer science, and artificial intelligence. Its development can be broadly divided into three eras:

The Industrial Robot Era (1960s-1990s)

1961: Unimate became the first industrial robot deployed in production, working on General Motors' assembly line
1969: Victor Scheinman at Stanford designed the Stanford Arm, a pioneer in electric robotic arms
1970s-1980s: PUMA (Programmable Universal Machine for Assembly) robots gained widespread adoption in manufacturing
Characteristics: Fixed-base, pre-programmed trajectories, repetitive tasks (welding, painting, material handling)
Key algorithms: PTP (Point-to-Point) control, PID regulation, trajectory interpolation

The Mobile Robot Era (1990s-2010s)

1997: NASA's Sojourner rover successfully landed on Mars
2000s: The DARPA Grand Challenge accelerated autonomous driving technology
2004: iRobot Roomba launched the consumer service robot market
2005: Boston Dynamics' BigDog demonstrated dynamically balanced quadruped locomotion
Characteristics: Autonomous navigation, environmental perception, dynamic motion control
Key technologies: SLAM, path planning, visual odometry

The Intelligent Robot Era (2010s-present)

2013: Boston Dynamics Atlas showcased highly dynamic humanoid locomotion
2016: AlphaGo sparked an AI revolution; deep learning permeated every area of robotics
2020s: Large Language Models (LLMs) empower robots with natural language interaction and task planning
Characteristics: Learning and adaptation, multimodal perception, human-robot collaboration
Key technologies: Deep reinforcement learning, Sim-to-Real, Foundation Models

Non-Learning Methods

Although robot learning dominates current research, over the past few decades many robotic systems have achieved remarkable capabilities without relying on learning algorithms at all. Classical methods still hold irreplaceable advantages in reliability, interpretability, and safety.

Major Achievements of Classical Control Theory

Achievement	Core Methods	Year
Apollo 11 Moon Landing	Trajectory optimization + robust control	1969
Mars Rover Autonomous Navigation	Path planning + visual odometry	1997-present
Boston Dynamics Atlas	Trajectory optimization + MPC + whole-body control	2013-present
Industrial Robot Welding	Inverse kinematics + PID control	1970s-present
da Vinci Surgical Robot	Master-slave teleoperation + motion scaling	2000-present

Classical Methods vs. Learning Methods

Classical method strengths: High interpretability, clear safety guarantees, no large datasets required, excellent real-time performance
Learning method strengths: Strong adaptability, handles high-dimensional perception, manages scenarios that are difficult to model
Trend: Deep integration of both -- learning methods for perception and high-level decision-making, classical methods for low-level safety guarantees

Kinematics

Kinematics studies the geometric relationship between robot joint motions and end-effector pose, without considering forces or torques.

Forward Kinematics

Given all joint angles \(\mathbf{q} = [q_1, q_2, \ldots, q_n]^T\), compute the end-effector pose \(\mathbf{T}_{0n}\).

Denavit-Hartenberg (DH) Convention:

Each joint is described by four parameters:

Parameter	Meaning
\(\theta_i\)	Joint angle (rotation about \(z_{i-1}\))
\(d_i\)	Link offset (translation along \(z_{i-1}\))
\(a_i\)	Link length (translation along \(x_i\))
\(\alpha_i\)	Link twist (rotation about \(x_i\))

The homogeneous transformation matrix for each joint:

\[ A_i = \begin{bmatrix} \cos\theta_i & -\sin\theta_i \cos\alpha_i & \sin\theta_i \sin\alpha_i & a_i\cos\theta_i \\ \sin\theta_i & \cos\theta_i \cos\alpha_i & -\cos\theta_i \sin\alpha_i & a_i\sin\theta_i \\ 0 & \sin\alpha_i & \cos\alpha_i & d_i \\ 0 & 0 & 0 & 1 \end{bmatrix} \]

The complete forward kinematics transformation: \(T_{0n} = A_1 A_2 \cdots A_n\)

Inverse Kinematics

Given a desired end-effector pose \(\mathbf{T}_{target}\), find the joint angles \(\mathbf{q}\). This is a mapping from task space to joint space and is generally much harder than forward kinematics.

Analytical (Closed-Form) Methods:

Applicable to special structures (e.g., 6-DOF manipulators with spherical wrists)
Directly solve using geometric relationships and algebraic equations
Pros: Fast computation, high accuracy
Cons: Only applicable to specific configurations

Numerical Methods:

Jacobian iteration: Uses the Jacobian matrix \(J(\mathbf{q})\) to relate joint velocities to end-effector velocities

\[ \dot{\mathbf{x}} = J(\mathbf{q})\dot{\mathbf{q}} \]

Iterate \(\Delta \mathbf{q} = J^{\dagger} \Delta \mathbf{x}\) to approach the target (\(J^{\dagger}\) is the pseudoinverse)

CCD (Cyclic Coordinate Descent): Optimizes one joint at a time; commonly used in animation and gaming
FABRIK (Forward And Backward Reaching Inverse Kinematics): An efficient geometry-inspired algorithm

Dynamics

Dynamics studies the relationship between forces/torques and motion, forming the core foundation of robot control.

Lagrangian Formulation

Based on energy methods, define the Lagrangian \(L = T - V\) (kinetic energy minus potential energy):

\[ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = \tau_i \]

Rearranged into standard form:

\[ M(\mathbf{q})\ddot{\mathbf{q}} + C(\mathbf{q}, \dot{\mathbf{q}})\dot{\mathbf{q}} + G(\mathbf{q}) = \boldsymbol{\tau} \]

Where:

\(M(\mathbf{q})\): Mass/inertia matrix (positive definite and symmetric)
\(C(\mathbf{q}, \dot{\mathbf{q}})\): Coriolis and centrifugal force matrix
\(G(\mathbf{q})\): Gravity vector
\(\boldsymbol{\tau}\): Joint torque vector

Newton-Euler Formulation

A recursive method based on force/torque balance with \(O(n)\) computational complexity (n = number of joints):

Forward recursion (base to tip): Compute velocities and accelerations for each link
Backward recursion (tip to base): Compute forces and torques at each joint

Compared to the Lagrangian method, the Newton-Euler formulation is more commonly used in real-time control due to its lower computational complexity.

Dynamics Simulation

Commonly used robot dynamics simulators:

Simulator	Features	Use Cases
MuJoCo	Efficient contact dynamics, GPU acceleration	RL training, dexterous manipulation
PyBullet	Open-source, easy to use	Rapid prototyping
Isaac Sim	NVIDIA GPU acceleration, photorealistic rendering	Sim-to-Real
Gazebo	ROS integration, active community	Mobile robots
Drake	Optimization-centric, mathematically rigorous	Manipulation planning

Perception

SLAM (Simultaneous Localization and Mapping)

SLAM is a core problem in mobile robotics: simultaneously building a map and localizing within it in an unknown environment.

EKF-SLAM (Extended Kalman Filter SLAM):

Concatenates the robot pose and all landmark positions into a single large state vector
Uses the EKF for prediction and update steps
Drawback: \(O(n^2)\) computational complexity (n = number of landmarks); unsuitable for large-scale environments
State vector: \(\mathbf{x}_t = [\mathbf{r}_t, \mathbf{m}_1, \mathbf{m}_2, \ldots, \mathbf{m}_n]^T\)

FastSLAM (Particle Filter SLAM):

Uses particle filters to estimate the robot trajectory
Each particle maintains an independent landmark map (via EKF)
Leverages Rao-Blackwellization to reduce complexity
FastSLAM 2.0 incorporates the latest observation into the proposal distribution for improved efficiency

Visual SLAM: ORB-SLAM, LSD-SLAM, VINS-Mono, and others

Visual Odometry

Estimates camera (robot) motion by matching features across consecutive image frames:

Feature extraction (ORB, SIFT, SuperPoint)
Feature matching
Motion estimation (Essential matrix / PnP)
Local optimization (Bundle Adjustment)

Point Cloud Processing

3D point cloud registration: ICP (Iterative Closest Point) algorithm
Point cloud segmentation: Deep learning methods such as PointNet / PointNet++
3D object detection: VoxelNet, PointPillars
Sensor fusion: Multimodal fusion of LiDAR and cameras

Control

PID Control

The most classical feedback control method:

\[ u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt} \]

P (Proportional): Current error
I (Integral): Accumulated error (eliminates steady-state error)
D (Derivative): Rate of error change (predicts trends, suppresses oscillations)

Tuning methods: Ziegler-Nichols method, manual tuning, auto-tuning

LQR (Linear Quadratic Regulator)

For a linear system \(\dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u}\), minimize the quadratic cost function:

\[ J = \int_0^\infty (\mathbf{x}^T Q \mathbf{x} + \mathbf{u}^T R \mathbf{u}) dt \]

The optimal control law is \(\mathbf{u} = -K\mathbf{x}\), where \(K = R^{-1}B^T P\) and \(P\) is obtained by solving the Riccati equation.

Q matrix: Penalty weights on state deviation
R matrix: Penalty weights on control effort
LQR provides the optimal linear state-feedback control

MPC (Model Predictive Control)

MPC solves a finite-horizon optimal control problem at each time step:

\[ \min_{\mathbf{u}_{0:N-1}} \sum_{k=0}^{N-1} \ell(\mathbf{x}_k, \mathbf{u}_k) + V_f(\mathbf{x}_N) \]

\[ \text{s.t.} \quad \mathbf{x}_{k+1} = f(\mathbf{x}_k, \mathbf{u}_k), \quad \mathbf{x}_k \in \mathcal{X}, \quad \mathbf{u}_k \in \mathcal{U} \]

Core advantages:

Naturally handles constraints (joint limits, torque limits, obstacle avoidance)
Can handle nonlinear systems
Receding-horizon optimization provides a degree of robustness

Common solvers: OSQP, IPOPT, acados, CasADi

Planning

Path Planning

Algorithm	Type	Features
A*	Graph search	Optimality guarantee, requires discretization
D / D Lite	Incremental search	Suitable for dynamic environments, supports replanning
PRM	Sampling-based	Efficient for multi-query, expensive preprocessing
RRT	Sampling-based	Efficient for single-query, friendly to high-dimensional spaces
RRT*	Sampling-based	Asymptotically optimal, slower convergence

Trajectory Optimization

Trajectory optimization generates smooth motion trajectories while satisfying dynamics and constraints:

CHOMP (Covariant Hamiltonian Optimization for Motion Planning): Gradient-based trajectory optimization using the covariant gradient of the cost function
STOMP (Stochastic Trajectory Optimization for Motion Planning): Gradient-free; explores the trajectory space through stochastic sampling
TrajOpt: Based on Sequential Convex Optimization
iLQR / DDP: Iterative optimization methods grounded in dynamics models

Intersection with AI Planning

Cross-References

Robot planning has close ties to AI planning. For search algorithms, heuristic methods, and decision planning in AI, see:

Classical AI search algorithms (A*, Monte Carlo Tree Search)
Planning in reinforcement learning (Model-Based RL, MCTS)
LLM-powered task planning (SayCan, Code as Policies)

Modern robot planning increasingly incorporates learning methods:

Movement Primitives: DMP, ProMP
Learned cost functions: Obtain planning cost functions through learning from demonstrations or inverse RL
End-to-end planning: Neural network planners mapping directly from perception to action
LLM + Planning: Use large language models for high-level task decomposition while retaining classical motion planning at the low level

References

Siciliano et al., Robotics: Modelling, Planning and Control, Springer
Lynch & Park, Modern Robotics: Mechanics, Planning, and Control, Cambridge University Press
Thrun, Burgard & Fox, Probabilistic Robotics, MIT Press
LaValle, Planning Algorithms, Cambridge University Press