Video 2.1a Vijay Kumar and Ani Hsieh

Robo3x-1.3 1 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Introduction to Lagrangian

Vijay Kumar and Ani Hsieh University of Pennsylvania

Robo3x-1.3 2 Property of Penn Engineering, Vijay Kumar and Ani Hsieh

• Aristotle • Euler • Galileo • Lagrange • Bernoulli • D’Alembert

1. Principle of Virtual : Static equilibrium of a particle, system of N particles, rigid bodies, system of rigid bodies 2. D’Alembert’s Principle: Incorporate inertial for dynamic analysis 3. Lagrange’s Equations of

Robo3x-1.3 3 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Generalized Coordinate(s) A minimal set of coordinates required to describe the configuration of a system

No. of = no. of degrees of freedom

Robo3x-1.3 4 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Displacements Virtual displacements are small displacements consistent with the constraints

g y

A particle of m constrained to move vertically generalized

coordinate y Robo3x-1.3 5 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Displacements Virtual displacements are small displacements consistent with the constraints l

f q

x

The slider crank : a single degree of freedom linkage

generalized coordinate x, q, or f Robo3x-1.3 6 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Displacements Virtual displacements are small displacements consistent with the constraints

O

q l

P The : a single degree of freedom linkage

generalized coordinate q Robo3x-1.3 7 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Work The work done by applied (external) forces through the virtual ,

mg

A particle of mass m constrained to move vertically

Robo3x-1.3 8 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Virtual Work The work done by applied (external) forces through the virtual displacement,

l

f q F

x

The slider crank mechanism: a single degree of freedom linkage

Robo3x-1.3 9 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.1b Vijay Kumar and Ani Hsieh

Robo3x-1.3 10 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium

mg

Robo3x-1.3 13 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium

l

f q F

x

Robo3x-1.3 14 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium

O

q l P mg

Robo3x-1.3 15 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s Principle

The virtual work done by all applied (external) forces through any virtual displacement is zero include inertial principle of virtual work forces

The system is in static equilibrium for the system inertial = - mass x

Robo3x-1.3 16 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s Principle

acceleration

inertial force e2

mg e1

Robo3x-1.3 17 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s Principle

acceleration O

q inertial force l e P q

mg er equation of motion

Robo3x-1.3 18 Property of Penn Engineering, Vijay Kumar and Ani Hsieh D’Alembert’s principle with generalized coordinates

generalized coordinate

contribution contribution from from inertial external force(s) force(s) Particle in the vertical plane

Simple pendulum

Robo3x-1.3 19 Property of Penn Engineering, Vijay Kumar and Ani Hsieh The Key Idea

The contribution from the inertial forces can be expressed as a function of the kinetic and its derivatives

Kinetic Energy

Robo3x-1.3 20 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Lagrange’s Equation of Motion

Particle in the vertical plane

Simple pendulum

Robo3x-1.3 21 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 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 Robo3x-1.3 22 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Standard Form of Lagrange’s Equation of Motion

The Lagrangian

Robo3x-1.3 23 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Lagrange’s Equation of Motion

Particle in the vertical plane

Simple pendulum

Robo3x-1.3 24 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.2 Vijay Kumar and Ani Hsieh

Robo3x-1.3 25 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations

Vijay Kumar and Ani Hsieh University of Pennsylvania

Robo3x-1.3 26 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations Recall Newton’s 2nd law of motion

m

net external force = x acceleration Robo3x-1.3 28 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations Newton’s equations of motion for a translating

Euler’s equations of motion for a rotating rigid body

Robo3x-1.3 29 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Motion of Systems of Particles

fj Center of Mass y Pi m i C

Pi fj rOP mj i x O z

Newton’s equations of motion net external force = total mass x

Robo3x-1.3 30 accelerationProperty of Penn Engineering, of Vijay Kumarcenter and Ani Hsieh of mass 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

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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 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 x

Robo3x-1.3 32 Property of Pennmoment Engineering, Vijay of Kumar inertia and Ani Hsieh 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 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 acting (with the same origin) on the body.

Robo3x-1.3 33 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Example

O

q l e2 P mg e1 equation of motion

Robo3x-1.3 34 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 67 KB

Moment of Inertia of Planar Objects

e2 m 2b m

2a C e O 1 m m

Robo3x-1.3 35 Property of Penn Engineering, Vijay Kumar and Ani Hsieh of Planar Objects

dm r

C

y

x

Robo3x-1.3 36 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.3 Vijay Kumar and Ani Hsieh

Robo3x-1.3 37 Property of Penn Engineering, Vijay Kumar and Ani Hsieh and Control Acceleration Analysis: Revisited

Vijay Kumar and Ani Hsieh University of Pennsylvania

Robo3x-1.3 38 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b2 Position Vectors

b1 • Reference frame Q b3 P • Origin B • Basis vectors • Position Vectors

• Position vectors for P and Q in A a a2 1 O

• Position vector of Q in B A

a3

Robo3x-1.3 39 Property of Penn Engineering, Vijay Kumar and Ani Hsieh General Approach to Analyzing Multi-Body System

2. Pair of points fixed to the same 1. Points common to body adjacent bodies

3. Pair of points on adjacent bodies whose relative can be easily described

4. Write equations relating pairs of points either on same or adjacent bodies.

Robo3x-1.3 40 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b2 Position Analysis

b1 Q

b3 P B

a a2 1 O

A

a3

Robo3x-1.3 41 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b Analysis 2

b1 Q

b3 P B

3x3 skew symmetric matrix a a2 1 O

A

a3

Robo3x-1.3 42 Propertyangular of Penn Engineering, velocity Vijay Kumar and of Ani HsiehB in A Acceleration Analysis

• Acceleration of P and Q in A Q

P B

a a2 1 O

A

a3

Robo3x-1.3 43 Property of Penn Engineering, Vijay Kumar and Ani Hsieh

The angular acceleration of B in Q A, is defined as the derivative of P the of B in A: B

a a2 1 O

A

a3

Robo3x-1.3 44 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b Acceleration Analysis 2

b1 Q

b3 P B

a a1 tangential 2 acceleration O

A

a3 centripetal (normal) acceleration Robo3x-1.3 45 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Serial Chain of Rigid Bodies

OD C OC D B

OB A OA

Robo3x-1.3 46 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Serial Chain of Rigid Bodies

OD C OC D B

OB A OA

Robo3x-1.3 47 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Video 2.4 Vijay Kumar and Ani Hsieh

Robo3x-1.3 48 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Newton Euler Equations (continued)

Vijay Kumar and Ani Hsieh University of Pennsylvania

Robo3x-1.3 49 Property of Penn Engineering, Vijay Kumar and Ani Hsieh 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 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Moment of Inertia of 3-D Objects?

dm r

C

y

x

Robo3x-1.3 51 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Inertia Tensor of 3-D Objects

dm r

C

y

x

Robo3x-1.3 52 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Inertia Dyadic

Principal Products of Moments of Inertia Inertia

Robo3x-1.3 53 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Example: Rectangular Plate Rotating about Axis through Center of Mass

A B B C C

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

A B B C C

Is the angular momentum parallel to the Robo3x-1.3 55 Property of Pennangular Engineering, Vijay velocity? Kumar and Ani Hsieh 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, uT I u, is called a principal moment of inertia.

Robo3x-1.3 56 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Examples

b2 b2

b1 b1

b3 b3

Robo3x-1.3 57 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Euler’s Equations of Motion

b2 In frame B

b C 1

b3

Robo3x-1.3 58 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Euler’s Equations of Motion need to differentiate in A b2

In frame A

b C 1

b3

Robo3x-1.3 59 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Euler’s Equations of Motion

Transform back to frame B b2

b C 1

b3 angular velocity in frame B

Robo3x-1.3 60 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Euler’s Equations of Motion

1. frame B (bi) along principal axes b2 2. center of mass as origin

3. all components along bi b C 1

b3

angular velocity, moments of inertia, moments in frame B Robo3x-1.3 61 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Quadrotor

w3

w2

w4

w1

mg

Robo3x-1.3 62 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Static Equilibrium (Hover)

w3

w2

w4

w1 Motor

Motor mg

Robo3x-1.3 63 Property of Penn Engineering, Vijay Kumar and Ani Hsieh F3

M3 F2 F 4 w3 M M 2 4 r w 3 F 2 r 1 w 1 4 M C 1 r w 4 r1 1 Resultant Force mg

Resultant Moment about C

Robo3x-1.3 64 Property of Penn Engineering, Vijay Kumar and Ani Hsieh b 3 Newton-Euler Equations B b2

Rotation of thrust b1 vector from B to A

Components in the inertial frame along a1, a2, and a3

u1

Components in the body frame along b1, b2, u and b3, the principal axesRobo3x-1.3 65 Property of Penn Engineering, Vijay2 Kumar and Ani Hsieh