Video 2.1a Vijay Kumar and Ani Hsieh

Video 2.1a Vijay Kumar and Ani Hsieh Robo3x-1.3 1

Introduction to Lagrangian Mechanics Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2

Analytical Mechanics Aristotle Galileo Bernoulli Euler Lagrange D Alembert 1. Principle of Virtual Work: Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies 2. D Alembert s Principle: Incorporate inertial forces for dynamic analysis 3. Lagrange s Equations of Motion Robo3x-1.3 3

Generalized Coordinate(s) A minimal set of coordinates required to describe the configuration of a system No. of generalized coordinates = no. of degrees of freedom Robo3x-1.3 4

Virtual Displacements Virtual displacements are small displacements consistent with the constraints y g A particle of mass m constrained to move vertically generalized coordinate y Robo3x-1.3 5

Virtual Displacements Virtual displacements are small displacements consistent with the constraints l q f x The slider crank mechanism: a single degree of freedom linkage generalized coordinate x, q, or f Robo3x-1.3 6

Virtual Displacements Virtual displacements are small displacements consistent with the constraints O q l The pendulum: a single degree of freedom linkage P generalized coordinate q Robo3x-1.3 7

Virtual Work The work done by applied (external) forces through the virtual displacement, mg A particle of mass m constrained to move vertically Robo3x-1.3 8

Virtual Work The work done by applied (external) forces through the virtual displacement, l q F f x The slider crank mechanism: a single degree of freedom linkage Robo3x-1.3 9

Video 2.1b Vijay Kumar and Ani Hsieh Robo3x-1.3 10

Virtual Work The work done by applied (external) forces through the virtual displacement, O q l P mg The pendulum: a single degree of freedom linkage Robo3x-1.3 11

The Principle of Virtual Work The virtual work done by all applied (external) forces through any virtual displacement is zero The system is in equilibrium Robo3x-1.3 12

Static Equilibrium mg Robo3x-1.3 13

Static Equilibrium l q F f x Robo3x-1.3 14

Static Equilibrium O q l P mg Robo3x-1.3 15

D Alembert s Principle The virtual work done by all applied (external) forces through any virtual displacement is zero include inertial forces principle of virtual work The system is in static equilibrium Equations of motion for the system inertial force = - mass x acceleration Robo3x-1.3 16

D Alembert s Principle acceleration inertial force e 2 mg e 1 Robo3x-1.3 17

D Alembert s Principle O acceleration q P l e q inertial force mg e r equation of motion Robo3x-1.3 18

D Alembert s principle with generalized coordinates generalized coordinate contribution from external force(s) Particle in the vertical plane contribution from inertial force(s) Simple pendulum Robo3x-1.3 19

The Key Idea The contribution from the inertial forces can be expressed as a function of the kinetic energy and its derivatives Kinetic Energy Robo3x-1.3 20

Lagrange s Equation of Motion Particle in the vertical plane Simple pendulum Robo3x-1.3 21

Lagrange s Equation of Motion for a Conservative System Conservative System There exists a scalar function such that all applied forces are given by the gradient of the potential function Scalar Function Gravitational force is conservative Scalar function is the potential energy Robo3x-1.3 22

Standard Form of Lagrange s Equation of Motion The Lagrangian Robo3x-1.3 23

Lagrange s Equation of Motion Particle in the vertical plane Simple pendulum Robo3x-1.3 24

Video 2.2 Vijay Kumar and Ani Hsieh Robo3x-1.3 25

Newton Euler Equations Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 26

Analytical Mechanics 1. Principle of Virtual Work: Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies 2. D Alembert s Principle: Incorporate inertial forces for dynamic analysis 3. Lagrange s Equations of Motion Robo3x-1.3 27

Newton Euler Equations Recall Newton s 2 nd law of motion m net external force = inertia x acceleration Robo3x-1.3 28

Newton Euler Equations Newton s equations of motion for a translating rigid body Euler s equations of motion for a rotating rigid body Robo3x-1.3 29

Motion of Systems of Particles Center of Mass y P i m i f j C P i f j z O r OPi m j x Newton s equations of motion net external force = total mass x acceleration Property of Penn Engineering, of Vijay Kumar center and Ani Hsieh of mass Robo3x-1.3 30

Newton s Second Law for a System of Particles The center of mass for a system of particles, S, accelerates in an inertial frame, A, as if it were a single particle with mass m (equal to the total mass of the system) acted upon by a force equal to the net external force. Linear momentum Rate of change of linear momentum in an inertial frame is equal to the net external force acting on the system. Also true for a rigid body Robo3x-1.3 31

Equations of Motion for a Rotating Rigid Body Rigid body with a point O fixed in an inertial frame Rigid body with the center of mass C Angular momentum O C y P moment of inertia x Robo3x-1.3 32

Equations of Motion for a Rotating Rigid Body Rigid body with a point O fixed in an inertial frame Rigid body with the center of mass C The rate of change of angular momentum of a rigid body (or a system of rigidly connected particles) in an inertial frame with O or C as an origin is equal to the net external moment acting (with the same origin) on the body. Robo3x-1.3 33

Example O e 2 e 1 q l P mg equation of motion Robo3x-1.3 34

67 KB Moment of Inertia of Planar Objects e 2 m 2b m 2a C Om m e 1 Robo3x-1.3 35

Moment of Inertia of Planar Objects r C dm y x Robo3x-1.3 36

Video 2.3 Vijay Kumar and Ani Hsieh Robo3x-1.3 37

Dynamics and Control Acceleration Analysis: Revisited Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 38

Position Vectors b 2 b 1 Reference frame Origin P b 3 B Q Basis vectors Position Vectors Position vectors for P and Q in A a 2 a 1 O Position vector of Q in B A a 3 Robo3x-1.3 39

General Approach to Analyzing Multi-Body System 2. Pair of points fixed to the same body 1. Points common to adjacent bodies 3. Pair of points on adjacent bodies whose relative motions can be easily described 4. Write equations relating pairs of points either on same or adjacent bodies. Robo3x-1.3 40

Position Analysis b 2 P b 3 B b 1 Q a 2 a 1 O A a 3 Robo3x-1.3 41

Velocity Analysis b 2 P b 3 B b 1 Q 3x3 skew symmetric matrix a 2 a 1 O A a 3 angular velocity of B in A Robo3x-1.3 42

Acceleration Analysis Acceleration of P and Q in A Q P B a 2 a 1 O A a 3 Robo3x-1.3 43

Angular Acceleration The angular acceleration of B in A, is defined as the derivative of the angular velocity of B in A: P B Q a 2 a 1 O A a 3 Robo3x-1.3 44

Acceleration Analysis b 2 P b 3 B b 1 Q tangential acceleration O a 2 a 1 A centripetal (normal) acceleration a 3 Robo3x-1.3 45

Serial Chain of Rigid Bodies O C B C O D D A O B O A Robo3x-1.3 46

Serial Chain of Rigid Bodies O C B C O D D A O B O A Robo3x-1.3 47

Video 2.4 Vijay Kumar and Ani Hsieh Robo3x-1.3 48

Newton Euler Equations (continued) Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 49

Newton Euler Equations Newton s equations of motion A rigid body B accelerates in an inertial frame A as if it were a single particle with the same mass m (equal to the total mass of the system) acted upon by a force equal to the net external force. Euler s equations of motion The rate of change of angular momentum of the rigid body B with the center of mass C as the origin in A is equal to the resultant moment of all external forces acting on the body with C as the origin Robo3x-1.3 50

Moment of Inertia of 3-D Objects? r C dm y x Robo3x-1.3 51

Inertia Tensor of 3-D Objects r C dm y x Robo3x-1.3 52

Inertia Dyadic Products of Inertia Principal Moments of Inertia Robo3x-1.3 53

Example: Rectangular Plate Rotating about Axis through Center of Mass A C B B C Is the angular momentum parallel to the Property of Penn angular Engineering, Vijay velocity? Kumar and Ani Hsieh Robo3x-1.3 54

Example: Rectangular Plate Rotating about Axis through Center of Mass A C B B C Is the angular momentum parallel to the Property of Penn angular Engineering, Vijay velocity? Kumar and Ani Hsieh Robo3x-1.3 55

Principal Axes and Principal Moments Principal axis u is a unit vector along a principal axis if I u is parallel to u There are 3 independent principal axes! Principal moment of inertia The moment of inertia with respect to a principal axis, u T I u, is called a principal moment of inertia. Robo3x-1.3 56

Examples b 2 b 2 b 1 b 1 b 3 b 3 Robo3x-1.3 57

Euler s Equations of Motion b 2 In frame B b 3 C b 1 Robo3x-1.3 58

Euler s Equations of Motion In frame A need to differentiate in A b 2 b 3 C b 1 Robo3x-1.3 59

Euler s Equations of Motion Transform back to frame B b 2 b 3 C b 1 angular velocity in frame B Robo3x-1.3 60

Euler s Equations of Motion 1. frame B (b i ) along principal axes b 2 2. center of mass as origin 3. all components along b i C b 1 b 3 angular velocity, moments of inertia, moments in frame B Robo3x-1.3 61

Quadrotor w 3 w 2 w 4 w 1 mg Robo3x-1.3 62

Static Equilibrium (Hover) w 3 w 2 w 4 Motor Speeds w 1 Motor Torques mg Robo3x-1.3 63

F 3 F 4 M 4 w 4 M 3 w 3 r 3 r 1 C r 4 r 1 Resultant Force mg F 1 F 2 M 2 w 2 M 1 w 1 Resultant Moment about C Robo3x-1.3 64

B b 3 Newton-Euler Equations b 2 b 1 Rotation of thrust vector from B to A Components in the inertial frame along a 1, a 2, and a 3 u 1 u 2 Components in the body frame along b 1, b 2, and b 3, the principal axes Robo3x-1.3 65