Manipulator Dynamics: Two Degrees-of-freedom

Transcription

1 Manipulator Dynamics: Two Degrees-of-freedom 2018 Max Donath

2 Manipulator Dynamics Objective: Calculate the torques necessary to overcome dynamic effects Consider 2 dimensional example Based on Lagrangian approach The Lagrangian: Joseph-Louis Lagrange ( ) Generalized coordinate Torque or force

3 For Mass M 1 For mass M 2, the Cartesian coordinates are The velocities are The velocity squared

4 Thus: Note: And the kinetic energy The Lagrangian is then

5 Differentiating the Lagrangian for T 1 Thus on joint 1, the torque is

6 Differentiating the Lagrangian for T 2 The torque on joint 2 is then

7 Inertial or sensitivity matrix Centripetal matrix Effective inertias Coupling inertias D 12 = D 21 Centripetal forces acting on joint i due to a velocity at joint j D 111 = D 222 = 0 & D 122 = D 211 Coriolis forces acting on joint i due to velocities at joints j & k D 112 = D 121 & D 212 = D 211

8 Manipulator Dynamics: Multiple Degree-of-freedom Systems

9 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

10 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

11 Computation of Joint Torques Fastest Approach: Table look-up approach, but large memory 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 manipulator on PDP11/45 with floating point: Assembly language: 4.5 msec FORTRAN: 33.5 msec (too slow)

12 Inertial effects of drive and transmissions on dynamics

13 Effect of Activator/Drive on Inertial Effects: (Ignoring compliance and dissipative losses)

14 J L Equiv = Equivalent inertia of load & motor/transmission on load shaft Typically Thus (Speed Reduction & Torque Increase) 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 at some configuration As the inertial effect increases the natural frequency decreases (as the square root)

15 However Note that as is reduced For: - Speed Reduction - Torque Increase - Reduction of loading variations (T L ) applied to the motor The power requirement for a given w L increases dramatically Power required for motor= P = T m w m when therefore This limits the minimum can be selected ratio that

16 Harmonic Drives (brand name of strain wave gear) See : 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.

17 HARMONIC DRIVE DEFINITIONS Cup sets consist of three components a flexspline, circular spline and a wave generator bearing. Flexspline A non-rigid or flexible, thin-walled, cylindrical cup with external teeth. It has two less teeth than the Circular Spline Circular Spline A 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 Generator The 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

18 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

19 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.

20 (Play video from separate file) 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. The UR5 uses harmonic drives. See

21 Harmonic Drive Reduction Ratio For a strain wave gearing mechanism, the gearing reduction ratio can be calculated from the number of teeth on each gear: Reduction ratio = Flex spline teeth circular spline teeth Flex spline teeth For example, if there are 202 teeth on the circular spline and 200 on the flex spline, the reduction ratio is ( )/200 = 0.01 From

22 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 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

23 How it Works The Cyclo speed reducing system is based on 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, see from Sumitomo from Nabtesco, which competes with Sumitomo See

24 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 see (adapted from Sumitomo literature) and see animation at

25 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: