Name: Robotics Perception, Planning and Control: Mathematical Foundations and Practical Implementation of SLAM, Path Planning and Motion Control
Author: step-3.5-flash

Introduction to Robotics: Core Concepts and Prerequisites

Over the past two decades, robotics has evolved from repetitive fixed-base industrial automation to dynamic, environment-interacting mobile and embodied systems, driven by breakthroughs in sensing, computing, and artificial intelligence. For intermediate practitioners who already grasp basic linear algebra and programming, this book focuses on the three core pillars of autonomous robot operation: perception, planning, and control, with a special emphasis on mobile robots that navigate unstructured real-world environments. As the opening chapter, this section first establishes a unified classification framework for robotics to clarify the scope of our discussion, then reviews the prerequisite mathematical and conceptual foundations required for subsequent chapters, and finally provides hands-on exercises to validate your knowledge baseline.

Robotics Classification Framework Robotics systems can be categorized along three primary dimensions: application scenario, locomotion type, and level of autonomy. Understanding these categories is critical to contextualizing the algorithms and challenges discussed throughout this book.

The first broad category is fixed-base robots, the most mature and widely deployed class of robotics systems. Fixed-base robots are anchored to a static surface and operate within a limited, predefined workspace. The most common subcategory is industrial manipulators, 6-degree-of-freedom (DOF) or 7-DOF robotic arms used for welding, assembly, and material handling. Leading commercial examples include the Universal Robots UR5 collaborative arm, KUKA LBR iiwa, and FANUC M-710 series. These systems rely on precise, pre-programmed trajectories and high-accuracy joint encoders, with minimal need for environmental perception or online planning. A second subcategory of fixed-base robots includes SCARA (Selective Compliance Articulated Robot Arm) systems, optimized for high-speed pick-and-place tasks in electronics manufacturing, and delta robots, used for high-throughput packaging and food processing.

The second and primary focus of this book is mobile robots, systems that can navigate uncontrolled physical environments without fixed anchoring. Mobile robots are further subdivided by locomotion type:

Wheeled robots: The most common class for logistics, delivery, and passenger transport. Subtypes include differential drive robots (e.g., iRobot Roomba, TurtleBot), omnidirectional wheeled robots (e.g., Amazon Robotics AMR), and Ackermann-steering autonomous vehicles (e.g., Tesla Autopilot-enabled cars, Waymo robotaxis).
Legged robots: Designed for unstructured terrain traversal, with leading examples including Boston Dynamics Spot quadruped and Atlas humanoid, as well as Tesla Optimus and Figure 01 humanoids for embodied AI research.
Aerial robots: Unmanned aerial vehicles (UAVs) such as DJI Mavic drones, used for aerial surveying, delivery, and search-and-rescue.
Underwater robots: Autonomous underwater vehicles (AUVs) and remotely operated vehicles (ROVs) used for ocean exploration and pipeline inspection.

The third category is special-purpose robots, built for niche high-stakes applications. This includes surgical robots such as the da Vinci Surgical System, space robots such as NASA’s Perseverance Mars rover, and hazardous environment robots such as bomb disposal units and nuclear inspection robots.

A key distinction between fixed-base and mobile robots lies in their constraint profiles: fixed-base manipulators are subject only to joint limits and collision constraints, while mobile robots are subject to non-holonomic constraints imposed by their locomotion systems. For example, an Ackermann-steering car cannot move laterally without forward motion, and a differential drive robot cannot rotate in place while translating linearly. These constraints fundamentally shape the design of planning and control algorithms for mobile systems, a theme we will revisit throughout this book.

```mermaid graph TD A[Robotics] --> B[Fixed-Base Robots] A --> C[Mobile Robots] A --> D[Special-Purpose Robots] B --> B1[Industrial Manipulators (e.g., UR5, KUKA LBR)] B --> B2[SCARA & Delta Robots] C --> C1[Wheeled Robots (AMR, Autonomous Vehicles)] C --> C2[Legged Robots (Boston Dynamics Atlas, Spot)] C --> C3[Aerial Robots (Drones, UAVs)] C --> C4[Underwater Robots (AUVs, ROVs)] D --> D1[Surgical Robots (da Vinci System)] D --> D2[Space Robots (Perseverance Rover)] D --> D3[Service Robots (Cleaning, Delivery)] ```

Core Problem Domains for Mobile Robots For mobile robots operating in unstructured environments, autonomous operation is decomposed into three tightly coupled modules: perception, planning, and control. Unlike fixed-base robots that follow pre-defined trajectories, mobile robots must continuously sense their environment, reason about their state and goals, and adjust their motion to account for uncertainty and dynamic changes.

The perception module answers two fundamental questions: *Where am I?* (state estimation) and *What is around me?* (environment perception). State estimation fuses data from on-board sensors such as IMUs, wheel encoders, LiDARs, and cameras to compute the robot’s 6-DOF pose (position and orientation) in a global reference frame, while compensating for sensor noise and drift. Environment perception detects and classifies obstacles, identifies traversable terrain, and builds a map of the surrounding area. For example, an autonomous vehicle uses camera and LiDAR data to detect pedestrians, traffic signs, and lane markings, while a legged robot uses depth cameras to identify flat, stable terrain for foot placement.

The planning module answers two complementary questions: *Where do I want to go?* (global planning) and *How do I avoid collisions in the short term?* (local planning). Global planning computes a collision-free path from the robot’s current position to a user-specified goal, using a pre-built or incrementally constructed map of the environment. For example, a warehouse AMR uses a global planner to compute a path from the charging station to a picking location, while a delivery drone uses a global planner to compute a route that avoids no-fly zones and buildings. Local planning runs at 10-100 Hz to adjust the robot’s trajectory in real time to avoid dynamic obstacles (e.g., a pedestrian walking into the robot’s path) and account for localization uncertainty.

The control module answers the question: *How do I execute the planned trajectory accurately?* The low-level motion controller takes the desired trajectory from the planning module and computes actuator commands (e.g., wheel velocities, joint torques) to track the trajectory as closely as possible, while compensating for model uncertainties, external disturbances (e.g., wind for drones, slippery ground for legged robots), and actuator latency. For example, a self-driving car’s lateral controller adjusts the steering angle to keep the vehicle centered in its lane, while a humanoid robot’s whole-body controller adjusts joint torques to maintain balance while walking over uneven terrain.

These three modules form a closed loop, as shown in the diagram below. Perception provides the state estimate and environmental data required for planning, planning generates the reference trajectory for the controller, and the controller’s execution produces new sensor data that feeds back into the perception module.

```mermaid graph LR A[Sensors (LiDAR, IMU, Camera)] --> B[Perception Module (State Estimation, Object Detection)] B --> C[Planning Module (Global Path Planning, Local Obstacle Avoidance)] C --> D[Control Module (Trajectory Tracking, Motion Regulation)] D --> E[Robot Actuation] E --> A ```

Mobile robots face three unique challenges that are rarely encountered in fixed-base systems: 1. Non-holonomic constraints: As noted earlie