Manipulator Dynamics: Two Degrees-of-freedom

©2017 Max Donath

Manipulator Dynamics

Objective: Calculate the torques necessary to overcome dynamic effects

Consider 2 dimensional example

Based on Lagrangian approach

The Lagrangian:

Generalized coordinateTorque or force

… Joseph-Louis Lagrange (1736-1813)

For Mass M1

For mass M2, the Cartesian coordinates are

The velocities are

The velocity squared

Thus:

Note:

And the kinetic energy

The Lagrangian is then

Differentiating the Lagrangian for T1

Thus on joint 1, the torque is

Differentiating the Lagrangian for T2

The torque on joint 2 is then

Inertial or sensitivity matrix

Centripetal matrix

Effective inertias

Coupling inertias D12 = D21

Centripetal forces acting on joint i due to a velocity at joint j

D111 = D222 = 0 & D122 = D211

Coriolis forces acting on joint i due to velocities at joints j & k

D112 = D121 & D212 = D211

Manipulator Dynamics:Multiple Degree-of-freedom Systems

In general, the dynamic equation for a manipulator in terms of a six dimensional displacement vector q

(6 joints, 7 links and a gripper) is

= 6 x 6 inertia matrix

= 6 x 6 viscous friction matrix

= 6 x 1 vector defining Coriolis and

centrifugal forces

= 6 x 1 vector defining the gravity terms

= 6 x 1 vector of input generalized forces

Bottleneck

Computation of joint torques in order to maintain joint positions, velocities, and accelerations

Variety of methods used, based on:

◆ Lagrangian techniques

◆ Newton-Euler Formulation

◆ Table look-up approaches

Computation of Joint Torques

Fastest Approach:

Table look-up approach, but largememory requirement

Most reasonable approach:

Iterative form of Newton Euler with dynamics referred to links’ internal coordinate system

(Luh, Walker & Paul, 1980)

In 1980, solution for 6 DOF manipulatoron PDP11/45 with floating point:

Assembly language: 4.5 msec FORTRAN: 33.5 msec (too slow)

Inertial effects of drive and transmissionson dynamics

Effect of Activator/Drive on Inertial Effects:

(Ignoring compliance and dissipative losses)

JL Equiv= Equivalent inertia of load & motor/transmission on load shaft

Typically

Thus may contribute a large constant inertia to the overall effective inertia … reducing the inertial variation somewhat.

Effect on Structural Natural Frequency:

Subscripts:

a: actual

o: measured atsome configuration

As the inertial effect increases the natural frequency decreases (as the square root)

(Speed Reduction & Torque Increase)

However

Note that as is reduced

For: - Speed Reduction

- Torque Increase

- Reduction of loading variations (TL) applied to the motor

The power requirement for a given wL increases dramatically

Power required for motor= P = Tmwm

therefore

This limits the minimum ratio that can be selected

when

Harmonic Drives◆ See http://www.harmonicdrive.net/

◆ 1955: Invented by C. Walton Musser◆ 1958: Harmonic Drive developed at United Shoe

Machinery Corp. Research Division. Patent issued◆ 1960: Harmonic Drive Div’n formed under United

Shoe Machinery, later changed to USM Corp.◆ 1991: Harmonic Drive Technologies acquired by

Teijin Seiki Ltd.◆ 2003: Teijin Seiki Ltd and Nabco Corporation Ltd

combine to form Nabtesco Corporation and Harmonic Drive Technologies, Teijin Seiki Boston Inc becomes Harmonic Drive Technologies, Nabtesco Incorporated

◆ 2006: HD Systems, Inc. and Harmonic Drive Technologies, Nabtesco Inc. formed joint company named Harmonic Drive LLC to market and support product.

HARMONIC DRIVE DEFINITIONS

◆ Cup sets consist of three components a flexspline, circular spline and a wave generator bearing.

◆ FlexsplineA non-rigid or flexible, thin-walled, cylindrical cup with external teeth. It has two less teeth than the Circular Spline

◆ Circular SplineA round, rigid ring with teeth on the internal spline that contact the teeth on the outside of the Flexspline. It is normally fixed but can be used as the output.

◆ Wave GeneratorThe Wave Generator is an elliptical ball bearing assembly. It normally functions as the input. It is inserted into the Flexspline causing the external teeth of the Flexspline to engage with the internal teeth of the Circular Spline at two equally spaced areas 180 degrees apart.

For video, see… http://www.youtube.com/watch?v=q1v7a0jsp1I

Elliptical wave generator(a ball bearing assembly) input deflects Flexspline to engage teeth at the major axis

Flexspline teeth at minor axis are fully disengaged – most of the relative motion occurs here

Flexspline output rotates in opposite direction to input

Rigid circular spline is rotationally fixed with a rigid internal gear

Deflections are greatly exaggerated

The teeth on the non-rigid Flexspline and the rigid Circular Spline are in continuous engagement.

Since the Flexspline has two teeth fewer than the Circular Spline, one revolution of the input causes relative motion between the Flexspline and the Circular Spline equal to two teeth.

With the Circular Spline rotationally fixed, the Flexspline rotates in the opposite direction to the input at a reduction ratio equal to one-half the number of teeth on the Flexspline.

All tabulated harmonic drive gear reduction ratios assume output through the Flexspline with the Circular spline rotationally fixed. However, any drive element may function as the input, output, or fixed member.

Natural preload provides zero backlash:

- Deflection of the round, free-state cup Flexspline into an elliptical shape at its open end results in non-parallel gear teeth along the major axis.- However, the teeth when engaged with the Circular Spline are returned to parallel, thereby creating a natural pre-load condition.- The result is minimal backlash.

https://www.youtube.com/watch?v=bzRh672peNk

(Play video from separate file)

Cyclo Drives

◆ Rolling gear action Planocentric Trochoidal (cycloidal) Gear Drives, or CYCLOs for short.

◆ Properties:– Large reduction in 1 stage – Smooth and high performance– High reliability, long service life– Excellent shock absorption, reduced friction– Compact and high power– Up to 500% overload capacity– Low to zero backlash.

◆ Lorenz Braren invented the Cyclo gearbox and founded Cyclo Getriebebau GmbH in Munich, Germany, in 1931.

◆ The Cyclo is a speed reducer without a high speed pinion or gear teeth; it does not operate in shear but in compression

Benefit: No catastrophic failure.◆ In 1974, Sumitomo Heavy Industries, acquired an

interest in Cyclo Getriebebau GmbH.◆ In 1994, Sumitomo bought the German company

outright.◆ See http://www.sumitomodrive.com/

How it Works

◆ The Cyclo speed reducing system is based on only three major moving parts:– High speed input shaft with an integral eccentric cam

and roller bearing assembly – Cycloid discs– Slow speed output shaft assembly

◆ As the input shaft and eccentric cam rotate, it rolls the cycloid discs around the internal circumference of the stationary ring gear.

◆ Resulting action is similar to that of a wheel rolling around the inside of a ring.

◆ As the cycloid disc travels in a clockwise path around the ring, the wheel itself turns slowly on its own axis in the opposite (counter clockwise) direction.

◆ Because of the curved, cycloidal profile of the cycloid disc, it progressively engages with the rollers of the ring gear housing to produce a reverse rotation at reduced speed.

◆ For each complete revolution of the high speed shaft, the cycloid disc turns one cycloidal "tooth" in the opposite direction.

◆ The reduced rotation of the cycloid disc is transmitted to the slow speed output shaft to provide - speed reduction.

◆ For videos on cyclo drives, seehttp://www.youtube.com/watch?v=_3OwTd0GG78

… from Sumitomohttps://www.youtube.com/watch?v=MBWkibie_5I

… from Nabtesco, which competes with SumitomoSee http://www.nabtescomotioncontrol.com

Why it's Better(from Cyclo literature)

◆ The smooth, progressive rolling action of the cycloidal disc speed reduction mechanism eliminates friction and the pressure points of conventional gears.

◆ All of the Cyclo's torque transmitting components roll, and are symmetrically placed around the shafts for balanced, smooth, vibration free and quiet operation.

◆ Because the components are in compression, they cannot shear off like traditional gear teeth.

Also seehttp://www.youtube.com/watch?v=EFni742pEXIAndhttps://www.youtube.com/watch?v=ie2wwAnKDM0

◆ Consider sample CYCLO of:- 18-roller ring gear (blue), two 17-lobe cycloid discs

(gray & black), an 8-roller planocentric ring gear or “spider”(green), and an eccentric shaft (cyan) with two eccentric portions (magenta) - shown at home position in the left layout in the box below.

◆ Each disc is supported radially by its own antifriction bearing (yellow) and eccentric.

◆ The right layout in the box shows the same unit at home position with the gray disc and the corresponding bearing and eccentric removed, for illustrating the 180 degree angle between the two eccentrics.

◆ The eccentric shaft, ring gear shaft, and spider shaft are coaxial.

Eccentric shaft: