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

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-acc n relationship As motion is linear only consider F

Example 1: 1 d.o.f. cart-spring Use Newtonian mechanics: Free body diagram:

Example 2: 2 d.o.f. cart-spring-pendulum Derive force-acc n and torque relationship Consider KE of cart and pendulum V pen = V cart + V pen/cart

Example 2: 2 d.o.f. cart-spring-pendulum PE = PE cart + PE pendulum

Example 2: 2 d.o.f. cart-spring-pendulum Find derivatives for linear & rotational motion See Niku, page 123

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! KE 2

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 θ 1 and θ 2 Get 2 equations of motion for T 1 and T 2 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

KE for link 1, V=0 : VD2VD2 KE KE = KE 1 + KE 2

PE = KE - PE Example 4: 2 d.o.f. robot arm

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 acc n at joint i Coupling inertia between joints i & j due to acc n 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) = 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 R T H = R T 1 1 T 2 ….. n-1 T H = A 1 A 2 …..A n For 6 d.o.f. robot: T 6 = 0 T 1 1 T 2 ….. 5 T 6 = A 1 A 2 …..A 6 Using the standard matrix for A

Derivative of matrix A (revolute) For a revolute joint w.r.t. its joint variable θ i

Derivative of matrix A - simplified Here matrix is broken into a “constant” matrix Q i and the (original) A i matrix so: QiQi

Derivative of matrix A (prismatic) Similarly QiQi

Q i matrix for both prismatic & revolute Both Q i matrices are always constant: Consider extending this for 0 T i matrix with multiple joint variables (i.e. θ ’s & d’s) which we are now calling q i

For multiple joints – derivative of 0 T i We need to find the partial differential of matrix 0 T i 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 0 T i for each link i. Only one Q j (select appropriately)

Example 5 Find an expression for the derivative of the transformation of the 5 th link of the Stanford Arm relative to the base frame w.r.t. 2 nd & 3 rd joint variables Stanford Arm is a spherical robot where 2 nd joint is revolute and the 3 rd joint is prismatic

Example 6 Higher order derivatives can be done:

Carry on next week To find KE, PE, Lagrangian and equation of motion