Robotics Notes

Robotics (INF-2600)

Introduction to AI in Robotics

• Robotics Definition: A multidisciplinary field combining mechanical engineering, electronics, computer science, and AI to create autonomous systems. These systems are designed to perceive, decide, and act within the physical world.
• Intelligence in Robotics: Robots exhibit intelligence when they can adapt, learn, and respond to sensory input rather than merely executing pre-programmed routines.

Embodied Intelligence

• Embodied AI: This concept shifts from software agents to physical robots, emphasizing intelligence expressed through a physical body that learns via interaction with the real world.
• Historical Context: Embodied AI's evolution can be traced from early robots like Shakey (1960s) to the advanced robots of Boston Dynamics.

Core Components of an AI Robot

• Perception:
  • Involves using sensors such as cameras, LiDAR, IMU (Inertial Measurement Unit), and touch sensors.
  • AI enhances perception through object detection, localization, and SLAM (Simultaneous Localization and Mapping).
• Reasoning and Decision Making:
  • Includes path planning and obstacle avoidance.
• Learning in Robotics:
  • Utilizes Reinforcement Learning (RL), imitation learning, and Sim2Real techniques.

Introduction to Manipulator Kinematics

• Inspired by Burton, York University

Robotic Manipulators

• Definition: A robotic manipulator is a kinematic chain, which is an assembly of rigid bodies connected by joints that allow movement relative to each other through mechanical constraints.
  • Links: The rigid bodies in the manipulator.
  • Joints: The mechanical constraints that allow relative movement between links.

Joints

• Types of Joints:
  1. Revolute (Rotary):
     • Functions like a hinge, allowing relative rotation about a fixed axis between two links.
     • By convention, the axis of rotation is the z-axis.
  2. Prismatic (Linear):
     • Functions like a piston, allowing relative translation along a fixed axis between two links.
     • By convention, the axis of translation is the z-axis.
• Joint Convention: Joint $i$ connects link $i-1$ to link $i$. When joint $i$ is actuated, link $i$ moves.

Joint Variables

• Degrees of Freedom (DOF): Revolute and prismatic joints are one degree of freedom joints, described by a single numeric value called a joint variable.
• Joint Variable Notation: $q_i$ represents the joint variable for joint $i$.
  1. Revolute:
     • $q<em>i = \theta</em>i$: Angle of rotation of link $i$ relative to link $i-1$.
  2. Prismatic:
     • $q<em>i = d</em>i$: Displacement of link $i$ relative to link $i-1$.

Revolute Joint Variable

• For a revolute joint, the joint variable $q<em>i$ is equal to the angle of rotation $\theta</em>i$ of link $i$ relative to link $i-1$.

Prismatic Joint Variable

• For a prismatic joint, the joint variable $q<em>i$ is equal to the displacement $d</em>i$ of link $i$ relative to link $i-1$.

Common Manipulator Arrangements

• Most industrial manipulators have six or fewer joints.
  • The first three joints form the arm.
  • The remaining joints form the wrist.
• Manipulators are often described using the joints of the arm:
  • R: Revolute joint
  • P: Prismatic joint

Articulated Manipulator

• Configuration: RRR (all three joints are revolute).
• Joint Axes:
  • $z_0$: Waist
  • $z<em>1$: Shoulder (perpendicular to $z</em>0$)
  • $z<em>2$: Elbow (parallel to $z</em>1$)

Spherical Manipulator

• Configuration: RRP
• Example: Stanford arm

SCARA Manipulator

• Configuration: RRP (revolute-revolute-prismatic)
• Full Name: Selective Compliant Articulated Robot for Assembly

Forward Kinematics

• Problem: Given the joint variables and link dimensions, determine the position and orientation of the end effector.

Forward Kinematics - Process

• Choose the base coordinate frame of the robot.
• Express the position $(x, y)$ in this frame.
• Link 1 moves in a circle centered on the base frame origin:
  $(a<em>1 \cos \theta</em>1 , a<em>1 \sin \theta</em>1)$
• Choose a coordinate frame with its origin located on joint 2, oriented the same as the base frame.
• Link 2 moves in a circle centered on frame 1:
  $(a<em>2 \cos(\theta</em>1 + \theta<em>2), a</em>2 \sin(\theta<em>1 + \theta</em>2))$
• Sum the coordinates because the base frame and frame 1 have the same orientation. The position of the end effector in the base frame is:
  $(a<em>1 \cos \theta</em>1 + a<em>2 \cos(\theta</em>1 + \theta<em>2), a</em>1 \sin \theta<em>1 + a</em>2 \sin(\theta<em>1 + \theta</em>2))$
• Determine the orientation of frame 2 with respect to the base frame, expressing $x<em>2$ and $y</em>2$ in terms of $x<em>0$ and $y</em>0$.
  • $x<em>2 = (\cos(\theta</em>1 + \theta<em>2), \sin(\theta</em>1 + \theta_2))$
  • $y<em>2 = (-\sin(\theta</em>1 + \theta<em>2), \cos(\theta</em>1 + \theta_2))$

Inverse Kinematics

• Problem: Given the position (and possibly the orientation) of the end effector and the link dimensions, determine the joint variables.
• Challenge: Inverse kinematics is more complex than forward kinematics because there is often more than one possible solution.

Inverse Kinematics - Law of Cosines

• Using the law of cosines:
  $b^2 = a<em>1^2 + a</em>2^2 - 2 a<em>1 a</em>2 \cos(\pi - \theta_2) = x^2 + y^2$
• Therefore:
  $\cos(\pi - \theta<em>2) = \frac{a</em>1^2 + a<em>2^2 - x^2 - y^2}{2 a</em>1 a_2}$
• And:
  $\cos(\theta<em>2) = \frac{x^2 + y^2 - a</em>1^2 - a<em>2^2}{2 a</em>1 a_2}$
• Let $C = \frac{x^2 + y^2 - a<em>1^2 - a</em>2^2}{2 a<em>1 a</em>2}$, then $\cos \theta_2 = C$
• Taking the inverse cosine gives only one of two possible solutions.
• Using trigonometric identities to obtain both solutions:
  $\sin^2 \theta + \cos^2 \theta = 1$
  $\tan \theta = \frac{\sin \theta}{\cos \theta}$
  $\theta = \tan^{-1} \frac{\pm \sqrt{1 - C^2}}{C}$
• Calculate $c<em>2$ and $s</em>2$:
  $c<em>2 = (x<em>x + y</em>y – a1<em>a1 – a2</em>a2) / (2<em>a1</em>a2)$
  $s</em>2 = \sqrt{1 – c2*c2}$
• Solutions for $\theta<em>2$: $\theta</em>{21} = \text{atan2}(s<em>2, c</em>2)$
  $\theta<em>{22} = \text{atan2}(-s</em>2, c_2)$

Spatial Descriptions

Points and Vectors

• Point: A location in space.
• Vector: Magnitude (length) and direction between two points.

Coordinate Frames

• Choosing a frame (a point and two perpendicular vectors of unit length) allows assignment of coordinates.

Dot Product

• Definition: The dot product of two vectors $u$ and $v$.
  $u = \begin{bmatrix} u<em>1 \ u</em>2 \ \vdots \ u<em>n \end{bmatrix}, v = \begin{bmatrix} v</em>1 \ v<em>2 \ \vdots \ v</em>n \end{bmatrix}$
  $u \cdot v = u<em>1 v</em>1 + u<em>2 v</em>2 + \dots + u<em>n v</em>n = \sum<em>{i=1}^{n} u</em>i v_i = u^T v$
  $u \cdot v = ||u|| ||v|| \cos \theta$

Translation

• Suppose we are given $o<em>1$ expressed in frame ${0}$: $^0o</em>1 = \begin{bmatrix} 3 \ 0 \end{bmatrix}$

Translation 1

• The location of frame ${1}$ expressed in frame ${0}$:
  $^0d<em>1 = ^0o</em>1 - ^0o_0 = \begin{bmatrix} 3 \ 0 \end{bmatrix} - \begin{bmatrix} 0 \ 0 \end{bmatrix} = \begin{bmatrix} 3 \ 0 \end{bmatrix}$

Translation 2

• $p<em>1$ expressed in frame ${0}$. $^0p = ^0d</em>1 + ^1p = \begin{bmatrix} 3 \ 0 \end{bmatrix} + \begin{bmatrix} 1 \ 1 \end{bmatrix} = \begin{bmatrix} 4 \ 1 \end{bmatrix}$

Translation 3

• $q_0$ expressed in frame ${0}$.
  $^0q = ^0p + ^0d = \begin{bmatrix} -1 \ 1 \end{bmatrix} + \begin{bmatrix} 3 \ 0 \end{bmatrix} = \begin{bmatrix} 2 \ 1 \end{bmatrix}$

Rotation

• Suppose that frame ${1}$ is rotated relative to frame ${0}$ by an angle $\theta$.

Rotation 1

• The orientation of frame ${1}$ expressed in ${0}$. The rotation matrix $^0R<em>1$ can be interpreted as the orientation of frame ${j}$ expressed in frame ${i}$. $^0R</em>1 = \begin{bmatrix} ^0x<em>1 \cdot ^1x & ^0y</em>1 \cdot ^1x \ ^0x<em>1 \cdot ^1y & ^0y</em>1 \cdot ^1y \end{bmatrix}$

Rotation 2

• $p<em>1$ expressed in frame ${0}$. The rotation matrix $^0R</em>1$ can be interpreted as a coordinate transformation of a point from frame ${j}$ to frame ${i}$.
  $^0p = ^0R_1 ^1p = \begin{bmatrix} \cos \theta &amp; - \sin \theta \ \sin \theta &amp; \cos \theta \end{bmatrix} \begin{bmatrix} 1 \ 1 \end{bmatrix}$

Rotation 3

• $q_0$ expressed in frame ${0}$.

  $^0q = ^0R_1 ^0p = \begin{bmatrix} \cos \theta &amp; - \sin \theta \ \sin \theta &amp; \cos \theta \end{bmatrix} \begin{bmatrix} -1 \ 1 \end{bmatrix}$

Properties of Rotation Matrices

• $R^T = R^{-1}$
• The columns of $R$ are mutually orthogonal.
• Each column of $R$ is a unit vector.
• $\det R = 1$ (the determinant is equal to 1).

Rotations in 3D

• Rotation matrix in 3D:

$^0R<em>1 = \begin{bmatrix} ^0x</em>1 \cdot ^1x &amp; ^0y<em>1 \cdot ^1x & ^0z</em>1 \cdot ^1x \ ^0x<em>1 \cdot ^1y & ^0y</em>1 \cdot ^1y &amp; ^0z<em>1 \cdot ^1y \ ^0x</em>1 \cdot ^1z &amp; ^0y<em>1 \cdot ^1z & ^0z</em>1 \cdot ^1z \end{bmatrix}$

Reinforcement Learning for Control

• RL Concepts: State, Action, Reward, Policy, Q-value.
• Deep RL in Robotics: DDPG (Deep Deterministic Policy Gradient), PPO, SAC (Soft Actor-Critic).
• Control Types: Model-based vs. Model-free control.
• Case Studies: OpenAI Gym, MuJoCo, Robosuite.
• Challenges: Sample inefficiency, safety, real-time learning.

Human-Robot Interaction and Explainable Robotics

• Human-in-the-loop Systems: Robots that take feedback from humans (e.g., RLHF).
• Emotion and Intention Recognition: Essential for social and assistive robots.
• Explainability and Trust in Robotics: Important in healthcare, military, and collaborative settings.
• Ethical Issues and Biases in Robotic Decisions: Bias in models, decision transparency, responsibility.

Applications and Case Studies

• Healthcare Robots: Surgical, eldercare.
• Industrial Automation: Robot arms, warehouses.
• Autonomous Vehicles and Drones.
• Humanoids and Social Robots: Pepper, Sophia.
• Extreme Environments: Space, disaster zones.

Frontiers and Research Challenges

• Sim2Real gap, continual learning, multi-agent collaboration
• Embodied intelligence and developmental robotics
• Open-ended learning and self-repairing robots
• AI + Robotics + Neuroscience

DARPA Urban Challenge

• Date: November 3, 2007
• Objective: Complete a 96 km urban area course in 6 hours.
• Challenge: Multiple robotic vehicles carry out missions simultaneously on the same course.
• Basic Rules:
  • Use a stock vehicle.
  • Obey California driving laws.
  • Operate entirely autonomously.
  • Avoid collisions with objects typical of an urban environment.
  • Operate in parking lots.
• DARPA supplied an environment map with information on lanes, lane markings, stop signs, parking lots, and special checkpoints.

Autonomous Car Driving Sensors

• Surround View
• Blind Spot Detection
• Traffic Sign Recognition
• Cross Traffic Alert
• Emergency Braking
• Adaptive Cruise Control
• Pedestrian Detection
• Park Assist
• Collision Avoidance
• Eye/Face Tracking
• Rear Collision Warning
• Lane Departure Warning
• Long-Range Radar
• LIDAR
• Camera
• Short/Med-Range Radar
• Ultrasound

Junior - Autonomous Vehicle

• Equipped with various sensors:
  • Velodyne laser
  • Riegl laser
  • Applanix INS
  • SICK LMS laser
  • IBEO laser
  • SICK LDLRS laser
  • BOSCH Radar