MSc: Fundamentals of Robot Control

Fundamentals of Robot Control

• Course name: Fundamentals of Robot Control
• Code discipline: F19
• Subject area:

Short Description

This course covers the following concepts: Introductory nonlinear control over dynamic systems with the focus on robotics; Stability, pros and cons of nonlinear control systems.

Prerequisites

Prerequisite subjects

• python,
• numpy library,
• Google Colab environment
• CSE201 — Mathematical Analysis I and CSE203 — Mathematical Analysis II]: integration and differentiation, exponentials, gradient.
• CSE202 — Analytical Geometry and Linear Algebra I
• CSE204 — Analytic Geometry And Linear Algebra II: matrix multiplication, eigenvector and eigenvalue.
• CSE205 — Differential Equations: state-space representation, homogeneous and nonhomogeneous equations, general solution of linear 1st and 2nd order ODEs, stability of solutions.
• Linear control theory: concepts of feedback, open- and closed-loop systems, linear controllers (PD, PID)

Course Topics

Course Sections and Topics

Section	Topics within the section
Motion. Kinematics. Dynamics.	Free body motion Manipulator position and orientation Homogeneous transformations Forward and inverse kinematics Kinetic and potential energy Euler-Lagrange equations
Linear systems. Stability	State-space equations Eigenvalues and eigenvectors Phase plane analysis Energy and stability Lyapunov’s direct method Lyapunov stability analysis
Feedback control systems	Feedback and building control loops Stabilization and trajectory tracking Linear regulators (P, PD, PID)
Feedback linearization	Joint-space inverse dynamics of serial manipulators Stabilization and trajectory tracking problems Input-state linearization
Robust control	Sliding modes in dynamic systems Robust control in scalar and matrix form Stability and tuning of robust controllers Control law smoothening.

Intended Learning Outcomes (ILOs)

What is the main purpose of this course?

Control theory is an integral part of modern robotics, and there is a high chance that most students majoring in Robotics would face the problems of controlling a physical plant (a robot, manipulator, drone, autonomous vehicle) in their research and graduation work as well as their professional careers. Therefore, the main purpose of this course is to prepare the students for solving practical control problems by teaching the most fundamental approaches of nonlinear control used in modern robotics applications.

ILOs defined at three levels

Level 1: What concepts should a student know/remember/explain?

By the end of the course, the students should be able to ...

• Basic structure of differential equations describing motion of robotic manipulators,
• Motivation behind and the basic structure of feedback control systems,
• How to find control system’s error dynamics and methods to analyze it,
• General structure of linear controllers (P, PD, PID),
• Physical motivation behind Lyapunov stability analysis,
• Basic structure of robust control system.

Level 2: What basic practical skills should a student be able to perform?

By the end of the course, the students should be able to ...

• Name the main sources of nonlinearities in physical systems,
• Explain the cons of applying PID controllers to nonlinear systems,
• Understand pros and cons of feedback linearization method,
• Name pros and cons of robust control approach,
• Numerically solve differential equations in MATLAB environment.

Level 3: What complex comprehensive skills should a student be able to apply in real-life scenarios?

By the end of the course, the students should be able to ...

• Know how to analyze stability of physical systems with Lyapunov direct method,
• Design feedback linearization systems,
• Synthesize robust control systems and tune them,
• Implement nonlinear control in MATLAB environment to simulate the behavior of multi-DOF robotic systems.

Grading

Course grading range

Grade	Range	Description of performance
A. Excellent	80-100	-
B. Good	60-79	-
C. Satisfactory	40-59	-
D. Poor	0-39	-

Course activities and grading breakdown

Activity Type	Percentage of the overall course grade
Labs/seminar classes	0
Interim performance assessment	60
Exams	40

Resources, literature and reference materials

Open access resources

• “Applied nonlinear control,” J.-J. Slotine & Weiping Li. Pearson, 1991.
• “Robotics: modelling, planning and control,” Bruno Siciliano, Lorenzo Sciavicco, Luigi Villani, and Giuseppe Oriolo. Springer Science & Business Media, 2010.
• “Robot modeling and control,” Mark Spong, Seth Hutchinson, M. Vidyasagar. John Wiley & Sons, 2006.
• “Modern Control Systems” (13th Edition), Richard Dorf & Robert H. Bishop. Pearson, 2016.
• “Modern Robotics: Mechanics, Planning, and Control,” Kevin Lynch, Frank Park. Cambridge University Press, 2017 (also, check out their video materials on YouTube).

Teaching Methodology: Methods, techniques, & activities

Activities and Teaching Methods

Activities within each section

Learning Activities	Section 1	Section 2	Section 3	Section 4	Section 5
Homework and group projects	1	1	0	1	1
Testing (written or computer based)	1	1	0	1	0
Discussions	1	1	1	1	1
Oral polls	0	1	1	1	1
Reports	0	0	0	0	1

Formative Assessment and Course Activities

Ongoing performance assessment

Section 1

Activity Type	Content	Is Graded?
Question	Given initial and final object position and orientations, obtain the corresponding transformation matrix.	1
Question	Find the Jacobian for a given manipulator.	1
Question	For a given differential equation describing a physical system, and for given kinetic (K) and potential energies (U) do the following: Show that there always exists a solution with respect to acceleration and that it is unique; Demonstrate that the system’s Hamiltonian H = K + U remains constant in the absence of external torques and forces; Show that the rate of change of the total energy equals instantaneous mechanical power.	1
Question	For a given manipulator with known Jacobian, do the following: Find kinetic and potential energies of the robot; Drive the Euler-Lagrange differential equation of motion.	1
Question	Do the following with MATLAB Robotics Toolbox: Compute basic rotation and homogeneous transformation matrices; Create a two-link manipulator and solve its forward kinematics; Simulate forward dynamics; Compute inertial, Coriolis and gravitational forces for the manipulator.	0
Question	Compute and analyze inertial tensor for a given robotic manipulator;	0
Question	For manipulators with different kinematic configurations, analytically derive their Jacobian matrices;	0
Question	Derive and analyze Euler-Lagrange equations describing dynamics of a given manipulator (required torques, singularities).	0

Section 2

Activity Type	Content	Is Graded?
Question	Convert given differential equation into state-space form	1
Question	Find eigenvalue for a given matrix	1
Question	Show that given a constant matrix ${\textstyle {\textbf {M}}}$ and any time-varying vector ${\textstyle {\textbf {x}}}$ , the time derivative of the scalar ${\textstyle \mathbf {x} ^{T}\mathbf {Mx} }$ can be written in the given form.	1
Question	For a given system of differential equations: Find equilibria points; Given a Lyapunov function, analyze system stability using Lyapunov’s direct method; Analyze system stability in a given range.	1
Question	For a given differential equation, find the values of coefficients ${\textstyle k_{1}}$ and ${\textstyle k_{2}}$ that would make it critically damped.	0
Question	Convert a given system of differential equations into state-space form.	0
Question	For an equation given in the state space form, write it as an ordinary differential equation for specified input and output variables.	0
Question	Analyze system’s stability given equation of its full energy.	0
Question	Apply Lyapunov’s direct method to analyze stability of a physical system.	0

Section 3

Activity Type	Content	Is Graded?
Question	Give an example of using feedback in activities of daily life	1
Question	Drive error dynamics equations for a given feedback control law	1
Question	What are the physical analogies for the individual terms of PD-controller?	1
Question	How to implement PD regulator in MATLAB software?	1
Question	Solve numerically in MATLAB a second-order ODE for the following controller types: P, PD, PID.	0
Question	Analyze stability of a given linear control system.	0
Question	How individual gains of PD and PID controllers affect transient and steady-state response?	0
Question	How does underestimation of system parameters affect performance of linear controllers and how to improve it?	0

Section 4

Activity Type	Content	Is Graded?
Question	For a given differential equation that describes pendulum dynamicsm do the following: Find control law transforming original dynamics into that of a linear mass-spring-damper system; Write position error dynamics for the designed control law (inverse dynamics); Repeat the previous steps if there are uncertainties in some of the system’s parameters.	1
Question	For a given general form of dynamics equation, demonstrate that the following control laws guarantee system stability.	1
Question	Simulate dynamics of given nonlinear system in MATLAB for given control law.	1
Question	Perform input-state linearization for a given system of differential equations.	1
Question	Find control law that linearizes a given differential equation.	0
Question	Analyze stability of given nonlinear systems and contribution of individual terms to system stability.	0
Question	Find general feedback linearization law for a given system of differential equations.	0
Question	Analyze stability and limitations of a given feedback linearization control law over a two-link manipulator.	0

Section 5

Activity Type	Content	Is Graded?
Question	Name uncertainties typical for mechanical systems	1
Question	Analyze system behavior in sliding mode	1
Question	Simulate system behavior with robust control in MATLAB	1
Question	Numerically implement control law smoothening for a robust controller.	1
Question	How to synthesize robust controller for a given range of system parameters’ deviations?	0
Question	Describe the effect of robust gain coefficient ${\textstyle k}$ on system stability, control output and resulting system behavior.	0
Question	Design robust controller for a dynamical system described by a given set of differential equations and implement it in MATLAB.	0
Question	Analyze stability of robust control system for a robotic manipulator with given equation of motion.	0

Final assessment

Section 1

1. What is a rotation matrix and what does it describe?
2. How to find a homogeneous transformation matrix? How is it different from rotation matrix?
3. What is manipulator Jacobian? How does it relate static forces and torques? How can one use the Jacobian to analyze manipulator singularities?
4. What is physical nature of the terms of Euler-Lagrange equations of robot motion?
5. What are the main properties of the basic terms of differential equations of motion (invertibility, positive definiteness, singularities, limits).

Section 2

1. Evaluate stability of a linear system whose dynamics is written in the state-space form.
2. What must be the properties of Lyapunov function candidate and its time derivative to confirm

Local stability, Global stability of the system.

1. How to analyze system’s stability based on its phase portrait (with examples)?
2. Describe physical motivation behind Lyapunov’s direct method and how it used to analyze stability of dynamical systems.

Section 3

1. What is the physical analog of PD-regulator in application to control over second-oder mechanical systems?
2. For a given system described by differential equations, design a linear control system and analyze its stability.
3. Describe pros and cons of linear controllers in application to nonlinear system control.

Section 4

1. What are the pros and cons of feedback linearization approach?
2. Provide examples of systems (differential equations) for which feedback linearization can result in infinite control effort.
3. What are the typical issues when applying feedback linearization approach to control over robotic manipulators?

Section 5

1. Describe typical sources of uncertainties and parameter deviations in models of physical and mechanical systems, their typical ranges and influence on system behavior.
2. Name pros and cons of robust control systems with and without control law smoothening.
3. How to tune robust controller terms to compensate for system uncertainties?

The retake exam

Section 1

Section 2

Section 3

Section 4

Section 5