Teaching

Learning Objectives

By the end of this course, students will be able to:

• formulate and analyze the kinematics of rigid body chains using screw theory, demonstrating understanding of underlying mathematical concepts (such as rotation matrices, quaternions, exponential coordinates, twists, and homogeneous transformations) and their interrelations;

• derive the dynamics of rigid body chains by applying fundamental mechanics concepts (d'Alembert's principle, Lagrangian mechanics) and using symbolic math programming tools;

• write algorithms for simulating and analyzing the kinematics and dynamics of articulated robots (forward and inverse kinematics, forward and inverse dynamics) while addressing challenges like singularities and redundancy;

• abstract robot control systems as linear time-invariant (LTI) systems and analyze and design their stability using classical control techniques (root-locus, Nyquist, Bode plots, PID control);

• formulate basic state estimators (Luenberger observer, Kalman filter, extended Kalman filter) and write state estimation algorithms; and

• apply advanced control strategies for robot systems (computed torque control, feedback linearization, operational space control, impedance control, linear quadratic regulator, model-predictive control) to solve related robot motion and force control problems.

Course Schedule

The course is structured in six main chapters, progressing from rigid body fundamentals through manipulator kinematics and dynamics to control:

Chapter 1: Rigid Body Motions

Lectures 1-5 | Fundamentals, tracking rigid bodies, and differential kinematics

• Concepts of space, inertial frame, rigid body, body frame

• Orientation representations: Euler angles, angle-axis, quaternions, exponential coordinates

• Homogeneous transformations, twists, screw motion

• Rotational velocity, rigid body velocity, adjoint transformation

• Wrenches and their transformation

Chapter 2: Manipulator Kinematics

Lectures 5-10 | Forward kinematics, inverse kinematics, and differential kinematics

• Product of exponentials formula, relation to Denavit-Hartenberg

• Manipulator workspace, Paden-Kahan subproblems

• Geometric manipulator Jacobian, numerical inverse kinematics

• End-effector forces and joint torques

• Redundant manipulators, task prioritization, Moore-Penrose pseudoinverse

• Singular configurations, manipulability ellipsoid, parallel manipulators

Chapter 3: Rigid Body Dynamics

Lectures 10-13 | Dynamics of constrained particles and rigid bodies

• Newton's laws, d'Alembert's principle, Lagrange equations

• Linear and angular momentum, inertia matrix, Newton-Euler equations

• Dynamic parameter identification, rigid body Lagrangian

Chapter 4: Manipulator Dynamics

Lectures 14-16 | Open-chain and constrained manipulator dynamics

• Lagrangian, equations of motion, joint friction

• Contact models, Pfaffian constraints

• Solving dynamics with Lagrangian multipliers

• Newton-Euler algorithm for inverse and forward dynamics

Chapter 5: Fundamentals of System Control

Lectures 17-21 | LTI systems, stability, and state estimation

• State space representation, transfer functions, Laplace transformation

• Loop transfer, root locus, Nyquist criterion, stability margins

• PID control design and tuning

• Observability, Luenberger observer

• Kalman filter, Bayes filters, recursive state estimation

Chapter 6: Manipulator Control

Lectures 22-27 | Motion control, force control, and optimization-based control

• Independent joint control, inverse dynamics control via feedback linearization

• Lyapunov stability

• Operational space control

• Impedance control, direct force control

• Linear quadratic regulators and model-predictive control

Learning Resources

The course content draws from three textbooks:

• A Mathematical Introduction to Robotic Manipulation

by Richard M. Murray, Zexiang Li, and S. Shankar Sastry (1994)

Focus: Kinematics and Dynamics

• Modern Robotics

by Kevin M. Lynch and Frank C. Park (2017)

Focus: Kinematics and Dynamics

• Feedback Systems: An Introduction for Scientists and Engineers

by Karl Johan Åström and Richard M. Murray (2010)

Focus: Control

Additional recommended textbooks include:

• Robotics: Modeling, Planning and Control by Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, and Giuseppe Oriolo (2009)

• Applied Dynamics: With Applications to Multibody and Mechatronic Systems by Francis C. Moon (1998)

• Control System Design: An Introduction to State-Space Methods by Bernard Friedland (2005)

Programming Tools

Students use either MATLAB or Python for coding challenges. Students are responsible for ensuring they have access to and proficiency with their chosen environment. Resources include official documentation, CMU library resources, and online learning platforms focusing on scientific computing and numerical methods.

Assessment

Students' comprehension of theoretical concepts and practical application skills are assessed through biweekly assignments. These assignments typically contain mixed theoretical problems and coding challenges. Example tasks include:

• translating fundamental kinematics concepts like rotation matrices and exponential coordinates to planar motion problems,

• writing forward dynamics algorithms for robot mechanisms and simulating their dynamics,

• estimating the state of a moving object from noisy IMU data.