manipulator dynamics: two...
TRANSCRIPT
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: