Autonomous Robotics: Kinematics, SLAM, Trajectory Planning & Foundation Models
Robotics represents the physical embodiment of Artificial Intelligence. Where pure software algorithms exist in abstract data structures, robots must close the sensorimotor loop in the messy, continuous, unpredictable physical world through real-time perception, kinematic planning, and torque actuation.
1. Mathematical Kinematics: Forward vs. Inverse Kinematics
Joint Angles → End-Effector Pose
Computes the 3D Cartesian position and orientation of the robot's gripper/tool (x, y, z, θ)given the vector of internal joint angles θ ∈ ℝ^n using Denavit-Hartenberg (D-H) homogeneous transformation matrices:
Target Pose → Required Joint Angles
Given a desired target Cartesian coordinate in space, calculates the required joint angles. Highly non-linear with multiple or infinite solutions, solved iteratively via the Jacobian matrix:
2. SLAM: Simultaneous Localization and Mapping
An autonomous robot placed at an unknown location in an unknown environment must concurrently build a map of its surroundings while estimating its own pose within that map. Solved mathematically via probabilistic state estimation:
Visual & LiDAR SLAM
Extracts 3D sparse point cloud landmarks from stereo cameras or 64-beam LiDAR scans, optimizing reprojection error.
Factor Graph Optimization
Frameworks like GTSAM and Cartographer construct non-linear factor graphs, resolving loop closures to eliminate drift.
RRT* Path Planning
Rapidly-exploring Random Trees asymptotically find the optimal collision-free trajectory through high-dimensional C-space.
3. The Next Frontier: Vision-Language-Action (VLA) Foundation Models
Modern robotics is moving beyond isolated heuristic control stacks. Vision-Language-Action (VLA) models (e.g., Google RT-2, Figure 01, Open-X Embodiment) co-train on web-scale text, internet vision datasets, and physical robotic trajectories:
This unlocks zero-shot physical affordance understanding (e.g. asking a humanoid robot "pick up the object that can slice an apple" results in autonomous identification and safe manipulation of a kitchen knife).
