Video 2.1a Vijay Kumar and Ani Hsieh
Robo3x-1.3 1 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Introduction to Lagrangian Mechanics
Vijay Kumar and Ani Hsieh University of Pennsylvania
Robo3x-1.3 2 Property of Penn Engineering, Vijay Kumar and Ani Hsieh Analytical Mechanics
• Aristotle • Euler • Galileo • Lagrange • Bernoulli • 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 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 generalized coordinates = 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 mass 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 mechanism: 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 pendulum: 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 displacement,
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 Equations of motion for the system inertial force = - mass x acceleration
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 energy 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 potential energy 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 = inertia 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 rigid body
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 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 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 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 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 Moment of Inertia 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 Dynamics 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 motions 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 Velocity 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 Angular Acceleration
The angular acceleration of B in Q A, is defined as the derivative of P the angular velocity 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 Speeds
Motor Torques 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