Fractal branching ultra-dexterous robots (Bush robots) #
NASA ACRP Quarterly Report, February 1998, Hans Moravec
ADVANCED CONCEPTS RESEARCH PROJECTS
PR-Number 10-86888 Appropriation: 806/70110
CMU Cooperative Agreement NCC7-7
December 1, 1997 - February 27, 1998
Fractal branching ultra-dexterous robots
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213
February 20, 1998
Table of Contents
Section 1: Joint Model 4
Section 2: Control Wire Stresses 5
Section 3: General Excursion to Force Relation 8
Section 4: Test Joint Mechanics 9
Section 5: Test Joint Electrical and Thermal Design 12
Section 6: Observations 13
L ( 31/2 + 32/2
+ 33/2 + ... + 3i/2
+ ... ) =
L / (sqrt(3) 1) = 1.366 L
At the ith level, each branch has cross-sectional area proportional to 3i, but there are 3i branches with that cross section, so the net area, given by the product of those two numbers, remains constant. Thus, with perfect scaling, the mass of tree carried above any given branch is no more than 1.366 times the mass of the branch itself.
For purposes of developing the a working joint, we can therefore simulate the loading of a complete bush by extending the branch an additional 1.366 times its basic length, also ensuring that the extension masses 1.366 times as much. This is a very modest requirement for mechanical design of a joint, smaller even than the safety margins of two or more that would normally be incorporated.
Our prototype single joint will thus probably appear as follows:
The diameter of nitinol wire will be chosen to be able to lift, when contracting to its natural length, at least twice the weight of the branch plus ballast. We will be using thin aluminum sections for the branches. Our largest branch may be about 10 cm in length, and should weigh only a fraction of a kilogram. Wire with diameter of 0.25 mm, which can lift about one kilogram, should be more than adequate. This same wire requires a restoring force of about 0.2 kg to stretch it when cold.
Higher, smaller, levels of the bush will use correspondingly thinner wire, the cross section reduced by a factor of 1/3 for each level, the diameter by 1/sqrt(3).
2: Control Wire Stresses
We analyse the loading on the control wires due to weight of the bush structure by considering a branch extended horizontally. In the following diagram, L is the distance from the branch base to the attachment point of the control wires, B is the distance from the branch pivot of the base of the wires, C is the distance of the folded bush center of gravity from the pivot. W is the weight of the full bush above the pivot. The tilt angle of the segment below the horizontal is designated by .
Neglecting the lower wire, note that the upper wire must exert a
torque about the pivot cancelling the torque induced by the weight
W. The torque caused by the weight has magnitude W
C cos . If the tension in the wire is
designated by T, the wire exerts torque
In modeling the folded upper layers of the bush, we had concluded that the subtree above the length L segment was equivalent to extending the segment by an amount 1.366 L. This makes the length C = 1.183 L. We had also noted in the last report that letting B = L/20 allows a +/ 30 degree deflection with 5% wire elongation. Choose this value for B. Then the torque equivalence becomes:
From which we find the ratio of T to W:
This produces the following plot of wire tension as a function of , showing a maximum force T of under 25 times W, even when the angle is brought beyond the 30° limit, to 90° vertical:
The previous analysis orients the base horizontally, which causes the gravity torque to fall off as the branch deviates from its central orientation. A worse case occurs if the base is rotated to keep the branch horizontal at all times:
In this case,
and the curve looks as follows, with T still staying below 30 W
when the deflection is within +/ 30° :
A factor still missing in the above analyses is the force needed to stretch the opposing wire as the supporting power wire contracts. A typical nitinol wire when cold can be stretched with about 1/6 (i.e. less than 17%) the force it exerts when hot. This is small enough to be absorbable in a safety factor of two added to the above analysis. Thus we plan to use nitinol wire whose contraction strength is 60 times the weight of the bushlet it moves. The cold strength of a stretched wire would, in that case, not be enough to support the weight of its bush, if the forces in the above analysis were reversed. In that case, a mechanical stop at 30 degree deviation at the pivot of the branch could provide the necessary support.
3: General Excursion to Force Relation
The previous section examined the stresses for a particular branch excursion, +/ 30 degrees, for which the necessary wire contraction force was equal to 30 times the subbush weight (or 60 times, with a 2x safety factor).
To see how the wire stress depends on the desired excursion, we note from the special analysis that the maximum stress occurs at the point of maximum stretch of the wire (i.e. the maximum value of angle ). If we choose length units to make the branch length L = 1, last quarters report gives the maximum excursion as:
where ER is (E2 1)/( E2 + 1 ), and E is the length ratio of expanded to contracted wires. Our nitinol documentation suggests E = 1.05 allows long-term operation.
We can rewrite the relation from the previous section to reveal the general form for the wire force as a function of angle:
where C = (1 + 1/(sqrt(3)-1))/2 = 1.183 is the relative position of the folded subbush center of gravity.
Substituting max for in the expression for T/W gives the general expression of maximum wire force as a function of anchor position B:
where ER is the ratio (E2 1)/(E2 + 1). This results in the following relationship between maximum wire force T / W and maximum angular excursion max.
4: Test Joint Mechanics
Weve constructed a one-axis test bush segment joint out of threaded metal spacers and a hinge. Its basic length is 9 cm. With additional spacers to simulate the upper part of the bush, its length grows to 22 cm. The combined weight of this assembly is 115 grams. We mounted two control wires, each with a lateral offset of 0.5 cm from the side of the branch, to provide about +/ 30° of deflection with 5% wire elongation. This setup requires a contraction strength of about 7,000 grams, to support the full weight laterally, with a safety factor of two. A typical nitinol alloy (for instance the one used in Flexinol Muscle Wire) exerts this force in a wire about 0.66 mm in diameter.
Wire Resistance Heating Contraction Cooling
Diameter (ohm/meter) Current Force Time
.025 mm 1770 20 ma 7 g 1.1 sec
.050 mm 510 50 ma 35 g 1.3 sec
.100 mm 150 180 ma 150 g 1.8 sec
.150 mm 50 400 ma 330 g 3.0 sec
.250 mm 20 1 A 930 g 6.7 sec
.350 mm 9.0 2 A 2 kg 11 sec
.500 mm 5.1 4 A 4 kg 19 sec
.660 mm 2.54 7 A 7 kg 30 sec
.750 mm 2.2 8 A 9 kg 36 sec
1.00 mm 1.5 15 A 15 kg 55 sec
The charging resistor should refill the capacitor in the 30 second cooling time of the wire. The RC time constant should thus be perhaps 10 seconds. Since C is 3,500 microfarads, R is chosen to be 3 Kohms.
The Nitinol wire is hard and resists deformation. We found it difficult to anchor: it tended to slip out of regular pressure clamps. We were successful with one technique, inserting the wire through a small hole drilled through the side of a threaded spacer. The slightly bent nitinol wire can be securely pinched by a screw in the spacer:
The wire also resists bends, so it was necessary to pivot the anchors so the released the wire very close to its intended path. Even so, the wire exhibits bowing, even under stress. In future we will fasten pivoting eyelets to both ends of each wire. It will then be much easier to make straight runs of wire of exactly the correct length. These eyelets will be held in place by a pivot pin, forming a hinge: