Robot Dynamics – Slide Set 10 ME 4135 R. R. Lindeke, Ph. D.

We will examine two approaches to this problem Euler – Lagrange Approach: – Develops a “Lagrangian Function” which relates Kinetic and Potential Energy of the manipulator, as it is moving, thus dealing with the manipulator “As a Whole” in building force/torque equations Newton – Euler Approach: – This approach works to separate the effects of each link on machine torques by writing down its motion in a separable linear and angular sense. However, due to the highly coupled motions in a robot, it requires a forward recursion through the entire manipulator for building velocity and acceleration models of a link followed by a backward recursion for force and torque on each link ‘in turn’

Euler – Lagrange approach Employs a Denavit-Hartenberg structural analysis to define “Generalized Coordinates” for the structural models of the machine. It provides good insight into controller design related to STATE SPACE It provides a closed form interpretation of the various components in the dynamic model: – Due to Inertia – Due to Gravitational Effects – Due to Friction (joint/link/driver) – Due to Coriolis Forces relating motion of one link to coupling effects of other links’ motions – Due to Centrifugal Forces that cause the link to have a tendency to ‘fly away’ due to coupling to neighboring links and its own motion

Newton-Euler Approach A ‘computationally more efficient’ approach to force/torque determination It starts at the “Base Space” and moves forward toward the “End Space” – computing trajectory, velocity and acceleration demands then Using this ‘forward velocity’ information the control computes forces and moments starting at the “End Space” and moving back to the “Base Space”

Defining the Manipulator Lagrangian:

Generalized Equation of Motion of the Manipulator:

Starting Generalized Equation Solution We’ll initially focus on the Kinetic energy term (the hard one!) Remembering from physics: K. Energy = ½ mV 2 Lets define the velocities for the Center of Mass of a Link K:

Rewriting the Kinetic Energy Term: Notice the separation in velocities! m K is Link Mass D K is a 3x3 Inertial Tensor of Link K about its center of mass expressed W.R.T. the base frame – This term characterizes mass distribution of a rigid object

Focusing on D K : Looking at a(ny) link

For this Link: D C is the Inertial Tensor About it Center of Mass In General:

Defining the terms: The Diagonal terms are the “General Moments of Inertia” of the link The three distinct off diagonal terms are the “Products of Inertia” If the axes used to define the pose of the center of mass are aligned with the x and z axes of the link defining frame (i) then the products of inertia are zero and the diagonal terms form the “Principal Moments of Inertia”

Continuing after this simplification:

If the Link is a Rectangular Rod (of uniform mass) : This is a reasonable approximation for many arm links!

If the Link is a Thin Cylindrical Shell of Radius r and length L:

We must now Transform each link’s D c D c (for each link) must be defined in the Base Space to be added to the Lagrangian Solution for kinetic energy: D K = [R 0 K D C (R 0 K ) T ] Here R 0 K is the rotational sub-matrix defining the Link frame K (at its end) in the base space – (hum, seems like the thinking using DH ideas as we built a jacobian!)

Defining the Kinetic Energy due to Rotation (contains D K )

Completing our models of Kinetic Energy: Remembering:

Velocity terms are from Jacobians: We will define the velocity terms as parts of a “slightly” modified Jacobian Matrix: A K is linear velocity effect B K is angular velocity effect  I is 1 for revolute, 0 for prismatic joint types Velocity Contributions of all links beyond K are ignored – K+1, K+2 etc

Focusing on : This is a generalized coordinate of the center of mass of a link It is given by: A Matrix that essentially strips off the bottom row of the solution Note: Minus Sign

Re-Writing K. Energy for the ARM

Factoring out the Joint Velocity Terms Simplifies to:

Building an Equation for Potential Energy: This is a weighted sum of the centers of mass of the links of the manipulator Generalized coordinate of centers of mass (from earlier)

Finally: The Manipulator Lagrangian: Which means:

Introducing a ‘Simplifying’ Term D(q): This is the Manipulator Inertial Tensor D(q) is an n x n matrix sized by the robot!

Lets define “Generalized Forces” We say that a generalized force is an residual force acting on a arm after kinetic and potential energy are removed!?!*! The generalized forces are connected to “Virtual Work” through “Virtual Displacements” Displacements that are done without the physical constraints of time

Generalized Forces on a Manipulator We will consider in detail two (of the readily identified three): Actuator Force (torque) → Frictional Effects → Tool Forces →

Considering Friction (in greater detail): Friction is a non-linear and complex force opposing manipulator motion It consists of 3 contributions: Viscous friction Dynamic friction Static friction

These can be (jointly) modeled Defining a Generalized Coefficient of Friction for a link: Coeff. of Viscous Friction Coeff. of Dynamic Friction Coeff. of Static Friction

Combining these components of Virtual Work:

Building a General L-E Dynamic Model But Remembering: Starting with this term

Partial of Lagrangian w.r.t. joint velocity It can be ‘shown’ that this term equals (remembering D(q) earlier):

Completing the 1 st Term: This is found to equal:

Completing this 1 st term of the L-E Dynamic Model:

Looking at the 2 nd Term: This term can be shown to be: Notice: i (!) not 1

Before Summarizing the L-E Dynamical Model we introduce: A Velocity Coupling Matrix (n x n) A ‘Gravity’ Loading Vector (nx1)

The L-E (Torque) Dynamical Model is: Inertial Forces Coriolis & Centrifugal Forces Gravitational Forces Frictional Forces