Lagrangian Mechanics

A short overview

Introduction

Previously studied Kinematics and differential motions of robots

Now Dynamic analysis Inertias, masses, accelerations, loads. ΣF = ma and ΣT = Iα Spec actuators to move robot’s links under

largest loads

Lagrangian Mechanics

Based on the differentiation of energy terms with respect to the system’s variables and time.

For complex systems better to use than Newtonian Mechanics

Lagrangian, L = K.E. – P.E.• K.E. Kinetic Energy,

• P.E. Potential Energy

Lagrangian Mechanics

For Linear: Where F = Σ external

forces for linear motion

For Rotational Where T = Σ torques for

rotational motion

θ and x are system variables

iii

iii

LL

tT

x

L

x

L

tF

Equations of motion V Dynamics

We use differential motions to find equations of motion

Using dynamic equations, it is virtually impossible to solve them to find equations of motion!

Need inertial loading even in space…• (What’s unusual with the following …)

3 Canadian robotic arms (working together as a team to inspect the Space Shuttle Discovery). (Photo: NASA)

Example 1: 1 d.o.f. cart-spring

Ignoring wheel inertia, derive force-accn relationship As motion is linear only consider F

kxxmFx

L

x

L

tF

kxx

Lxmxm

txm

x

L

kxxmPEKEL

kxPExmmvKE

and ,,

2

1

2

12

1 and

2

1

2

1

22

222

Example 1: 1 d.o.f. cart-spring

Use Newtonian mechanics:• Free body diagram:

kxxmFkxmaF

makxF

maF

Example 2: 2 d.o.f. cart-spring-pendulum

Derive force-accn and torque relationship Consider KE of cart and

pendulum Vpen = Vcart + Vpen/cart

222 sincos

ˆsinˆcosˆ

llxV

llx

pen

jii

Example 2: 2 d.o.f. cart-spring-pendulum

PE = PEcart + PEpendulum

cos22

1

2

1

sin2

1cos

2

12

1

222

221

2

2

2

2

21

xllmxmmKE

lmlxmKE

xmKE

pend

cart

cos12

12

2 glmkxPE

Example 2: 2 d.o.f. cart-spring-pendulum

Find derivatives for linear & rotational motion• See Niku, page 123

cos22

1

2

1 222

221 xllmxmmL cos1

2

12

2 glmkx

Example 3: 2 d.o.f. link mechanism

Similar to a robot Now have more

acceleration terms:• Linear

• Radial

• Centripetal

• Coriolis

Use same method as before

Datum for PE = 0

Example 3: 2 d.o.f. link mechanism KE link 1: KE link 2:

• Write position equations in terms of x & y

• Differentiate, square & add together!

21

2111 2

1 lmKE

KE2

Example 3: 2 d.o.f. link mechanism

Total KE

Total PE Lagrangian

• See Niku p125 for detail and derivatives of L

KE - PE

2

Example 3: 2 d.o.f. link mechanism

Differentiate the Lagrangian with respect to the two system variables θ1 and θ2

Get 2 equations of motion for T1 and T2

Put in matrix form:

Example 4: 2 d.o.f. robot arm

Similar to Ex. 3, but have:• a change in coordinate

frames

• links have Inertial masses

Same steps as Ex. 3

22

2

1

2

1 ImVKE

KE for link 1, V=0 :

VD2

KE

KE = KE1 + KE2

PE

= KE - PE

Example 4: 2 d.o.f. robot arm

iii

LL

tT

Again these can be written in matrix form.

(Taken from Niku, p127.)

Moments of Inertia

As you would expect by now, the answers to the last 4 examples can be written in some symbolic standardised form.

For the 2 d.o.f. system:

Effective inertia at joint i gives torque due to angular accn at joint i

Coupling inertia between joints i & j due to accn at joint i (or j) causes torque at j (or i)

Represent centripetal forces acting at joint i due to velocity at joint j

Represent Coriolis acceleration Represent gravity forces at joint i

Dynamic Equations for robots

Last example was a 2 d.o.f. robot, we can do this for a multiple d.o.f. robot:• Long & complicated…

• Calculate KE & PE of links and joints

• Find the Lagrangian

• Differentiate the Lagrangian equation with respect to the joint variables

Kinetic Energy (KE) – in 3D

Vector equation of KE of a rigid body =

V

Kinetic Energy (KE) – 2D KE of a rigid body in planar motion (i.e. in one plane) =

22

2

1

2

1 IV mKKE

Need expressions for velocity of a point (e.g. c of m G) along a rigid body as well as moments inertia

Use D-H representation

RTH = RT1 1T2 ….. n-1TH = A1 A2 …..An

For 6 d.o.f. robot: T6 = 0T1 1T2 ….. 5T6 = A1 A2 …..A6

Using the standard matrix for A

1000

0 iii

iiiiiii

iiiiiii

i dCS

SaSCCCS

CaSSCSC

A

Derivative of matrix A (revolute)

For a revolute joint w.r.t. its joint variable θi

Derivative of matrix A - simplified

Here matrix is broken into a “constant” matrix Qi and the (original) Ai matrix so:

Qi

Derivative of matrix A (prismatic)

Similarly

Qi

Qi matrix for both prismatic & revolute

Both Qi matrices are always constant:

Consider extending this for 0Ti matrix with multiple joint variables (i.e. θ’s & d’s) which we are now calling qi

For multiple joints – derivative of 0Ti

We need to find the partial differential of matrix 0Ti

w.r.t. each joint variable Use variable j to represent a particular joint variable. Need to consider each link in turn so we need to

differentiate 0Ti for each link i.

Only one Qj (select appropriately)

Example 5

Find an expression for the derivative of the transformation of the 5th link of the Stanford Arm relative to the base frame w.r.t. 2nd & 3rd joint variables

Stanford Arm is a spherical robot where 2nd joint is revolute and the 3rd joint is prismatic

Example 6

Higher order derivatives can be done:

j

i

kk

ijijk q

T

qq

UU

0

Carry on next week

To find KE, PE, Lagrangian and equation of motion