A first course in legged robotics. The course is organized in a few ways.

1. The “Modules” tab divides the course material into a broader outcome (e.g., passive dynamic walker, hopper control) and suggests topics to cover to learn the outcome. We recommend this approach to the beginner.
2. The “Download/Links” tab can be used to download all material offline and provides links to external YouTube videos.
3. The “All Materials” tab provides all lectures material as they were taught. This is ideal if the beginner wants a semester long learning experience.

About: The course emphasizes learning through extensive numerical simulations and animations. Mathematical modeling and theory is explained to build the necessary intuition to develop numeric code. MATLAB code provides examples on how to program the physics, develop controllers, and animate the system (see Downloads/Links: MATLAB links for tutorials). CoppeliaSim (free simulator for education) is used for modeling and control code is written in Lua. MATLAB 2021 with symbolic and optimization toolbox and CoppeliaSim v4.1.0 (July 21, 2020) were used.

Module 1: 2D Passive hopper (MATLAB)

Topic (Units)	Videos	Lecture Notes	Source Code	Description
1. Foward Kinematics	Forward Kinematics Video	Forward Kinemetics PDF	Lec Code 1	Representing the kinematics using homogenous rotaions and translations Forward kinematics gives the end-effector position for the given joint angles llustrated using a two-link planar manipulator.
3. Dynamics using Euler-Lagrange equations	3a. Dynamics using Euler-Lagrange Equations	Dynamics of Euler-Lagrange PDF	Lec Code 3	Algorithm for solving for equations of motion using Euler-Lagrange equations. This is illistrated on a simple projectile withdrag Basics of differentiation and chain rule. Writing symbolic code to derive the equations of motion. Demonstrate how to derive equations of motion of a planar double pendulum
	3b. Deriving Equations of Motion
	3c. Double Pendulum Simulation
7. Intro. to Hybrid Systems	Hybrid Systems Bouncing Ball Example	Hybrid Intro PDF	Lec 7 Code	Illustrate a hybrid system, a bouncing ball. We develop basic framework for simulating the system by coding one bounce and then repeated calls to one bounce Introduction to passive dynamic walker as a hybrid system. Terminology of hybrid systems such as Poincare map, fixed point, linearized stability of the fixed point, and a simple analytical example.
	Hybrid Systems Termonology	Hybrid Terminology PDF
8. Passive Dynamic Walkers	8a.  Passive Dynamic Walker (Pt1)	Passive Dynamic Walker PDF 1	Lec 8 Code	(Pt1- derivation) Intuition behind writing the equations of motion. Use of Euler-Lagrage equations to derive the equation for stance phase. (Pt2 - derivation) Use of Euler-Lagrage equations to derive the equations for heelstrike. Some intuition about how to simulate the system (Pt3- coding) Live coding of the equations to generate steady state gaits and analyze their linearized stability (see Terminology for hybrid systems)
	8b. Passive Dynamic Walker (Pt2)	Passive Dynamic Walker PDF 2
	8c. Passive Dynamic Walker (Pt3) Coding

Topic (Units)	Videos	Lecture Notes	Source Code	Description
1. Foward Kinematics	Forward Kinematics Video	Forward Kinemetics PDF	Lec Code 1	Representing the kinematics using homogenous rotaions and translations Forward kinematics gives the end-effector position for the given joint angles llustrated using a two-link planar manipulator.
3. Dynamics using Euler-Lagrange Equations	3a. Dynamics using Euler-Lagrange Equations	Dynamics of Euler-Lagrange PDF	Lec Code 3	Algorithm for solving for equations of motion using Euler-Lagrange equations. This is illistrated on a simple projectile withdrag Basics of differentiation and chain rule. Writing symbolic code to derive the equations of motion. Demonstrate how to derive equations of motion of a planar double pendulum
	3b. Deriving Equations of Motion
	3c. Double Pendulum Simulation
6. Jacobian Applications	Jacobian and its Applications	Jacobian PDF	Lec 6 Code	Basics of Jacobian. We show how to compute the Jacobian using symbolic computations and finite differences. We present two applications of Jacobian Find the velocity of points of interest and illustrated on planar double pendulum Find the static forces on a double pendulum.
7. Intro. to Hybrid Systems	Hybrid Systems Bouncing Ball Example	Hybrid Intro PDF	Lec 7 Code	Illustrate a hybrid system, a bouncing ball. We develop basic framework for simulating the system by coding one bounce and then repeated calls to one bounce Introduction to passive dynamic walker as a hybrid system. Terminology of hybrid systems such as Poincare map, fixed point, linearized stability of the fixed point, and a simple analytical example.
	Hybrid Systems Termonology	Hybrid Terminology PDF
8. Passive Dynamic Walkers	Passive Dynamic Walker (Pt1)	Passive Dynamic Walker PDF 1	Lec 8 Code	Intuition behind writing the equations of motion. Use of Euler-Lagrage equations to derive the equation for stance phase. Use of Euler-Lagrage equations to derive the equations for heelstrike. Some intuition about how to simulate the system.
	Passive Dynamic Walker (Pt2)	Passive Dynamic Walker PDF 2
15. Trajectory simulation	15b. Trajectory Generation	Trajactory Generations PDF	Lec 15b Code	Generating a reference trajectory using polynomials and given start and end conditions
17.2 Control Patitioning/Feedback Linearization	Control Partitioning (pt1)	Feedback Lineraization PDF 1	Lec 17b Code	This section introduces control partitioning or feedback linearization to track the reference motion. The key idea is to cancel the nonlinear dynamics and gravitational dynamics followed by wrapping a feedback controller The concepts are illustrated using simple MATLAB examples. Additional methods such as feed-forward, gravity + feedback, proportional-integral-derivative control are also introduced.
	Control Partitioning (pt2)	Feedback Lineraization PDF 2	Lec 17c Code
23. Legged mechanics for Walkers	Walker with Feedback Linerazation	Mechanics of Walkers PDF	Lec 23 Code	Since one of the two joints of the robt are unactuated, one can only usal partial feedback to control one joint The other joint is left free and is analzed using tools from hybrid systems (see Terminology for hybrid systems)
14b. Foot Placement control of Walker	Walker foot placement	Walker Foot Control PDF	Lec 14b Code	This section sketches some ideas on developing a foot placement control using a table lookup. No Code provided

Topic (Units)	Videos	Lecture Notes	Source Code	Description
26. Euler Parameters and Rotations	Rotations and Euler Parameters	3D rotations and Velocity PDF	Lec 26 Code	Introduction to 3D rotations using Euler Angles
27. Angular Velocity	3D anguar Velocity	3D angular Velocity PDF	N/A	Euler-Lagrange equations applied to a free falling block This includes derivations, simulations and animation for the falling block.
29. Zero Reference Model	Zero Reference Model for Kimetics	Zero Reference Model PDF	Lec 29 Code	Since Euler-Lagrange equations for simple systems get unwiedly it is recommended to these time consuming equations to C (called MEX files) and call them from MATLAB This section illustrates how to write MEX files and demonstrates derivation, simulation, and animation of a n-link pendulum where the number of links n can be set by the user
	MEX Files MATLAB to C++ Interface
8. Passive Dynamic Walkers	Passive Dynamic Walker (Pt1)	Passive Dynamic Walker PDF 1	Lec 8 Code	Intuition behind writing the equations of motion. Use of Euler-Lagrage equations to derive the equation for stance phase. Use of Euler-Lagrage equations to derive the equations for heelstrike. Some intuition about how to simulate the system.
	Passive Dynamic Walker (Pt2)	Passive Dynamic Walker PDF 2
15. Trajectory generation	15b. Trajectory Generation	Trajactory Generations PDF	Lec 15b Code	Generating a reference trajectory using polynomials and given start and end conditions
17.2 Control Patitioning/Feedback Linearization	Control Partitioning (pt1)	Feedback Lineraization PDF 1	Lec 17b Code	This section introduces control partitioning or feedback linearization to track the reference motion. The key idea is to cancel the nonlinear dynamics and gravitational dynamics followed by wrapping a feedback controller The concepts are illustrated using simple MATLAB examples. Additional methods such as feed-forward, gravity + feedback, proportional-integral-derivative control are also introduced.
	Control Partitioning (pt2)	Feedback Lineraization PDF 2	Lec 17c Code
30. Biped Control and Simulations	Humaniod Modeling and Control	Humanoid PDF	Lec 30 Code	Uses Euler-Lagrange equations to derive the equation Using trajectory generation to generate profiles for hip and knee. Use of partial feedback linearization to control the 3D model. Includes simulations, generation of MEX files, and animations.

Topic (Units)	Videos	Lecture Notes	Source Code	Description
4. Intro. to Coppelia Sim	Projectile using Euler-Lagrange	N/A	Lec 4 Code	Shows how to model and simulate a 2D projectile with drag in CoppeliaSim.
9. Rimless Wheel Modeling in CoppeliaSim	Rimless Wheel Modeling Video	N/A	Lec 9 Code	Shows different ways of modeling a 2D rimless wheel: Using two wheels attached side to side Using a spherical joint to constrain the wheel to 2D.
15. Two tricks for simulating planar systems in CoppeliaSim	15a. Two tricks for 2D legged simulation	Walker Simulations PDF	Lec 15a Code	This rimless model shown here has a torso. The torso can be controlled to a constant pitch angle to get the rimless wheel to move continuously. (See 5 Modeling and control in CoppeliaSim on how to do position, velocity, and torque control).

Topic (Units)	Videos	Lecture Notes	Source Code	Description
4. Intro to Coppelia Sim	Projectile using Euler-Lagrange	N/A	Lec 4 Code	Shows how to model and simulate a 2D projectile with drag in CoppeliaSim.
9. CoppeliaSim Rimless Wheel Modeling	Rimless Wheel Modeling Video	N/A	Lec 9 Code	Shows different ways of modeling a 2D rimless wheel: Using two wheels attached side to side Using a spherical joint to constrain the wheel to 2D.
15. Two tricks for simulating planar systems in coppeliaSim	15a. Two tricks for 2D legged simulation	Walker Simulations PDF	Lec 15a Code	This rimless model shown here has a torso. The torso can be controlled to a constant pitch angle to get the rimless wheel to move continuously. (See 5 Modeling and control in CoppeliaSim on how to do position, velocity, and torque control).
22. Finite State machine to Control Pendulum	Using State machine to Control Pendulum	N/A	Lec 22 Code	A finite state machine is a control architecture to develop controllers for walking machines. This example illustrates how to program a finite state machine to do a swing up and hold. Might be helful to review lecture 5.
24. Passive Dynamic Walker	Passive Dynamic Walker Video	N/A	Lec 24 Code	A model of passive walking with articulated knees (to get ground clearance) and hip spring to tune frequency of the swinging legs. The two tricks (see 15 a)are used to keep track of absolute angles and rotate gravity. A finite state machine is used to generate the simulation.

Topic (Units)	Videos	Lecture Notes	Source Code	Description
MATLAB Tutorials	MATLAB Basics	Basics PDF	N/A	MATLAB tutorials filled with helpful knowledge Gives basic understanding of MATLAB Animations and Scripts
	MATLAB Scrpits	Scripts PDF
	MATLAB Animations	Animations PDF
1. Foward Kinematics	Forward Kinematics Video	Forward Kinemetics PDF	Lec Code 1	Representing the kinematics using homogenous rotaions and translations Forward kinematics gives the end-effector position for the given joint angles llustrated using a two-link planar manipulator.
2. 2D Inverse Kinematics	Inverse Kinemetics Video	Inverse Kinemetics PDF	Lec Code 2	Inverse kinematics is solving for joint angles for a given position of the end-effector. This is harder because there may be a single, multiple or no solution to the inverse kinematics problem. Here we use root finding to solver for the inverse kinematics and illustrate it on a 2D planar manipulator.
3. Dynamics using Euler-Lagrange Equations	3a. Dynamics using Euler-Lagrange Equations	Dynamics of Euler-Lagrange PDF	Lec Code 3	Algorithm for solving for equations of motion using Euler-Lagrange equations. This is illistrated on a simple projectile withdrag Basics of differentiation and chain rule. Writing symbolic code to derive the equations of motion. Demonstrate how to derive equations of motion of a planar double pendulum
	3b. Deriving Equations of Motion
	3c. Double Pendulum Simulation
	3d. 1D projectile using Euler-Lagrange
4. Intro. to Coppelia Sim	Projectile using Euler-Lagrange	N/A	Lec 4 Code	Shows how to model and simulate a 2D projectile with drag in CoppeliaSim.
5. Modeling in CoppeliaSim	Modeling Simple Pendulum	N/A	Lec 5 Code	Modeling for a simple pendulum in CoppeliaSim Then we demonstrate how to do velocity, postion and torque using Lua scripts.
6. Jacobian Applications	Jacobian and its Applications	Jacobian PDF	Lec 6 Code	Basics of Jacobian. We show how to compute the Jacobian using symbolic computations and finite differences. We present two applications of Jacobian Find the velocity of points of interest and illustrated on planar double pendulum Find the static forces on a double pendulum.
7. Intro. to Hybrid Systems	Hybrid Systems Bouncing Ball Example	Hybrid Intro PDF	Lec 7 Code	Illustrate a hybrid system, a bouncing ball. We develop basic framework for simulating the system by coding one bounce and then repeated calls to one bounce Introduction to passive dynamic walker as a hybrid system. Terminology of hybrid systems such as Poincare map, fixed point, linearized stability of the fixed point, and a simple analytical example.
	Hybrid Systems Termonology	Hybrid Terminology PDF
8. Passive Dynamic Walkers	8a.  Passive Dynamic Walker (Pt1)	Passive Dynamic Walker PDF 1	Lec 8 Code	(Pt1- derivation) Intuition behind writing the equations of motion. Use of Euler-Lagrage equations to derive the equation for stance phase. (Pt2 - derivation) Use of Euler-Lagrage equations to derive the equations for heelstrike. Some intuition about how to simulate the system (Pt3- coding) Live coding of the equations to generate steady state gaits and analyze their linearized stability (see Terminology for hybrid systems)
	8b. Passive Dynamic Walker (Pt2)	Passive Dynamic Walker PDF 2
	8c. Passive Dynamic Walker (Pt3) Coding
9. Rimless Wheel Modeling in CoppeliaSim	Rimless Wheel Modeling Video	N/A	Lec 9 Code	Shows different ways of modeling a 2D rimless wheel: Using two wheels attached side to side Using a spherical joint to constrain the wheel to 2D.
10. Jacobian Application Review	Jacobian Review Problems	Jacobian Review PDF	N/A	We review Jacobians disscussed in Unit 6.
11. Closed Loop 5-link Chain Simuations	Dynamics simulation of clsoed 5 chain loops	Closed Chain Loops PDF	Lec 11.1 Code	We demonstrate how to derive equations of a closed-loop chain and simulate. We start off deriving equations of a 5-link pendulum and then add a constraint to create the closed loop chain. Next, we show how to model and simulate a closed-loop chain in Coppelia Sim
	5-link Chain in Coppelia Sim		Lec 11.2 Code
12. Hopping model: Spring Loaded Inverted Pendulum	Spring load inverted pendulum model	Hopping Model PDF	Lec 12 Code	Using Euler-Lagrange equations to derive the equations of motion. Details on simulation and animation. Generating steady state gaits and analyzing linear stability (see terminolgy for hybrid systems)
13. CoppeliaSim Custom PID Control	Custom PID Control Video	N/A	Lec 13 Code	We illustrate how to do a custom PID control. The pendulum is in torque control mode and the PID is on either the position or suitable cartesian coordinate.
14. Foot Placment Control	Foot Placement Control for Hopper	Hopper Foot Control PDF	Lec 14.1 Code	Foot placement control of hopper using Raibert’s control This section derives a simple control law using some intuition. MATLAB code shows how the controller works.
	Foot Placement Control for Walker	Walker Foot Control PDF	Lec 14.2 Code	This section sketches some ideas on developing a foot placement control using a table lookup. No code provided
15. CoppeliaSim Walker and Hopper Simulations	Two tricks for 2D legged simulation	Walker Simulations PDF	Lec 15a Code	Generating a reference trajectory using polynomials and given start and end conditions
	15b. Trajectory Generation	Trajactory Generations PDF	Lec 15b Code
16. Trajectory Optimization	Trajectory Optimization Video	Trajectory Optimization PDF	Lec 16 Code	Shows how to setup trajectory optimization using shooting and transcription method. The example problem was to get a car to travel from start to goal in minimum time.
17.2 Control Patitioning/Feedback Linearization	Control Partitioning (pt1)	Feedback Lineraization PDF 1	Lec 17b Code	This section introduces control partitioning or feedback linearization to track the reference motion. The key idea is to cancel the nonlinear dynamics and gravitational dynamics followed by wrapping a feedback controller The concepts are illustrated using simple MATLAB examples. Additional methods such as feed-forward, gravity + feedback, proportional-integral-derivative control are also introduced.
	Control Partitioning (pt2)	Feedback Lineraization PDF 2	Lec 17c Code
18. Debugging Trajectory Optimization	Debugging Video	Optimization PDF	N/A	Discusses some issues for performing trajectory optimization for projectile and legged systems.
19. Projectile Simulation: Calling CoppeliaSim from MATLAB	MATLAB Projectile Sim via CS Sim API	N/A	Lec 19 Code	We demonstrate how to control a projectile from MATLAB using the remote API This video is similar to Unit 4, but using Remote API instead of Regular API
20. Solving Periodic Gait Using Fmincon	Solving Periodic Gait	N/A	Lec 20 Code	This video shows how to solve for periodic gaits for a compass gait walker on level ground that is powered only by an ankle push-off.
21. Control Partitioning in CoppeliaSim	Control Partitioning/Feedback Linearization	N/A	Lec 21 Code	We use reflexxes motion library to generate a trajectory and then use control partitioning to track it.
22. Finite State machine to Control Pendulum	Using State machine to Control Pendulum	N/A	Lec 22 Code	A finite state machine is a control architecture to develop controllers for walking machines. This example illustrates how to program a finite state machine to do a swing up and hold. Might be helful to review lecture 5.
23. Control of  Walker	Walker control using Partial Feedback Linearization	Mechanics of Walkers PDF	Lec 23 Code	Since one of the two joints of the robot are unactuated, one can only usal partial feedback to control one joint The other joint is left free and is analzed using tools from hybrid systems (see Terminology for hybrid systems)
24. Passive Dynamic Walker	Passive Dynamic Walker Video	N/A	Lec 24 Code	A model of passive walking with articulated knees (to get ground clearance) and hip spring to tune frequency of the swinging legs. The two tricks (see 15 a)are used to keep track of absolute angles and rotate gravity. A finite state machine is used to generate the simulation.
25. Feedback Linearization	Feedback Linearization for Closed Chains	Closed Chain Modeling and Simulation Control	Lec 25 Code	We demonstrate feedback linearization of closed chains. We demonstrate 3 models, symmetric linkage, parallel linkage with two points of actuation.
26. Euler Parameters and Rotations	Rotations and Euler Parameters	3D rotations and Velocity PDF	Lec 26 Code	Introduction to 3D rotations using Euler Angles
27. 3D Angular Velocity	3D anguar velocity	3D angular Velocity PDF	N/A	Euler-Lagrange equations applied to a free falling block This includes derivations, simulations and animation for the falling block.
28. 3D Dynamics	Free Falling Block	3D Dynamics PDF	Lec 28 Code	Illustrates the use of Euler-Lagrange equations in 3D on a block falling under gravity
29. Zero Reference Model	Zero Reference Model for Kinmatics	Zero Reference Model PDF	Lec 29 Code	Since Euler-Lagrange equations for simple systems get unwiedly it is recommended to these time consuming equations to C (called MEX files) and call them from MATLAB This section illustrates how to write MEX files and demonstrates derivation, simulation, and animation of a n-link pendulum where the number of links n can be set by the user
	MEX Files MATLAB to C++ Interface
30. Biped Control and Simulations	Humanoid Modeling and Control	Humanoid PDF	Lec 30 Code	Uses Euler-Lagrange equations to derive the equation Using trajectory generation to generate profiles for hip and knee. Use of partial feedback linearization to control the 3D model. Includes simulations, generation of MEX files, and animations.