Fractal branching ultra-dexterous robots (Bush robots) #
NASA ACRP Quarterly Report, November 1997, Hans Moravec
ADVANCED CONCEPTS RESEARCH PROJECTS
PR-Number 10-86888 Appropriation: 806/70110
CMU Cooperative Agreement NCC7-7
September 1, 1997 - November 30, 1997
Fractal branching ultra-dexterous robots
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213
November 20, 1997
Table of Contents
Section 1: Shape-Memory Branch Design Considerations 4
Section 2: Simple Branch Fundamental Structure 4
Section 3: Electrical Actuation 6
Section 4: Mechanical Geometry 7
Section 5: Mass Loading 10
Section 6: Electrical and Thermal Considerations 11
Section 7: Construction 12
To simplify construction, while avoiding slippage of the actuating wires, we plan to make the connection between the lower and upper shafts with a threaded section on the lower shaft, and slotted in a + pattern. The upper shaft will have a mating threaded opening. The opposing X coils may actually be a single wire wrapped through the slots, held in place by tightening the screw connection of the two parts of the shaft. The Y coils will be held similarly. Here is a top view of the lower shaft, showing the threading of the X wires. The Y wires would be wrapped around the two remaining quadrants of the slotted section:
Besides providing a solid mechanical connection, the joint will provide the electrical return for the current that heats the Nitinol wires.
The X and Y axes of actuation operate almost independently. For convenience, the following presentation will concentrate on just one axis, without loss of generality. The situation is thus seen in two dimensional cross-section, and can be illustrated by two-dimensional diagrams like the following:
3: Electrical Actuation
In addition to simplified mechanics, our preliminary design has a maximally simple electrical arrangement. Though in future we expect to control bushes with computer-controlled switches, our preliminary design uses manual controls.
A nitinol wire reverts to its original length when its temperature is raised above a critical temperature (also called the Austenite start temperature). The temperature must not be raised too high, however, or the coil will anneal and acquire a new natural length. We ensure that the heating is within the correct range by providing a prearranged amount of energy, stored in a properly sized capacitor. The capacitor is discharged into the Nitinol spring, which, as the major resistance in the circuit, absorbs most of the energy. The temperature rise is set approximately by this energy divided by the mass and heat capacity of the spring. The following circuit achieves the basic aims:
The values of the capacitor C and supply voltage V are chosen so the VC is a few times the energy required to raise the temperature of the coil from room temperature to its critical temperature. The switch is double-throw with a third, spring-returned, center-off position. When the switch is momentarily operated in one direction or the other, the branch swings in the same direction. The charging resistor R is chosen so that the time constant of the RC circuit is similar to the cooling time constants of the coils. This ensures that the two coils are not simultaneously hot, and that a single coil is not overheated by repeated switch operations.
4: Mechanical Geometry
Cold Nitinol wires can elongate as much as 8%, but only for a small number of cycles before losing their natural length. The number of cycles grows to millions if the elongation is kept below about 5%. Greater relative elongations can be achieved by coiling the wire, as in a spring or on a drum. Uncoiled wire would be the most convenient actuator, but the small elongation puts severe constraints on the arm geometry. The constant dimensions of the geometry are illustrated in the following diagram:
R is the radius of the branch, including all sources of bulk, L is its length from the pivot base to the wire attachment point, B is the baseline of the wire anchor. Motion of the branch introduces two variables, wire length W and tilt angle theta :
W is related to theta by the relationship:
W2 = L2 + B2 + R2 + 2B ( L sin(theta) R cos(theta) )
By symmetry, for any constraint on the extent of W, q will vary between plus and minus some particular value, i.e. thetamin = thetamax. For example, if the constraint is a 5% elongation of the wire, then the maximum deflection is defined by:
W2max / W2min = 1.052 =
(L2 + B2 + R2 + 2B ( L sin(thetamax) R cos(thetamax) )) /
(L2 + B2 + R2 2B ( L sin(thetamax) + R cos(thetamax) ))
The solution of the expression for thetamax is greatly simplified by introducing the approximation R = 0, which still leaves the situation fairly realistic. Then:
W2max / W2min = 1.052 =
(L2 + B2 + 2BL sin(thetamax)) / (L2 + B2 2BL sin(thetamax))
This expression can now be solved for sin(thetamax) and thus thetamax:
thetamax = arcsin( ((1.052 1) (1 + (B/L)2)) / (2 B/L ( 1.052 + 1 )) )
The following graph of this formula shows the maximum deflection to each side for a 5% elongating wire, as L/B is varied from 1 to 41.
Reasonable deflections seem be achievable with simple wires, if the anchor points are placed very close to the pivot. At L/B = 41, it appears that even a maximum deflection of 90° is possible. The curve is slightly modified if the upper attachment point is offset by the radius R of the branch. To first order, the effect of R, which we eliminated in this approximation, is to displace the anchor points outward by an amount R, for a similar deflection. Thus it appears a good anchor uncoiled wire in the general case might be about R + L/20 from the pivot, giving about + 30° of deflection. The results also lead us to choose L as large as possible. Thus, the wires should be attached at the top end of the branch, not midway, as in our initial diagrams.
5: Mass loading
An original branch of our preferred (B=3, D=2) bush has three smaller branches, each 1/sqrt(3) the liner dimension of the parent. Each of those has three 1/3 the size of the original branch, for a total of 9. At the third level there are 27 each (1/3)3/2 the length, and so on. If carried to infinity, a the extended reach of a tree above a branch of length L would be:
Lupper = 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).
6: Electrical and Thermal Considerations
Nitinol alloys can be formulated within a wide range of transition temperatures. Low temperature alloys require less power to actuate, but take longer to cool down and recover, since heat flow to the environment is proportional to the temperature difference. Commonly used alloys have transitions at 70° C and 90° C, compared to typical ambient temperatures of 20° C to 30° C. We may choose higher temperature alloys in the interests of speed in future (replacing wire with ribbon is also an option), but in the present experimental phase, the lower temperature alloys are preferable.
A 0.25 mm 70° C wire takes about 7 seconds to cool in still air, but can be heated in about a millisecond by a current of one amp. A 10 cm length has a resistance of about 2 ohms.
We plan to use the driving circuit introduced in Section 3. To ensure adequate current averaging one ampere over a millisecond into the 2 ohms, the voltage V must exceed 2 volts. Five volts is convenient. To provide one ampere for a millisecond, the capacitor C must store at least one millicoulomb. At two volts, this would require about 500 microfarads. Thus we plan to use 5 volt, 500 microfarad capacitors (one for X direction, and one for Y) to power the 10 cm base of the bush.
The charging resistor should refill the capacitor in the 7 second cooling time of a wire. The RC time constant should thus be perhaps 3 seconds. Since C is 500 microfarads, suggesting R should be about 2 Kohms.
The cross-section of the wires in the higher levels of the tree drop by a factor of 1/3 each level, and their lengths drop a factor of 1/sqrt(3). Naively, the driving capacitances should drop by the 1/3 and the driving voltages by 1/sqrt(3) to compensate. In fact, continuous cooling incurred by the greater relative surface area of the thinner wires must be compensated by relative increases in V, which thus should fall more slowly than 1/sqrt(3) per level. The cooling time constants also fall from one level to the next, requiring adjustment of the charging time constants. The smaller capacitors at each level affect the charging time in the right direction, so perhaps the R values can be similar at each level.
We have obtained a large collection of Nitinol wire of various diameters and transition temperatures, and are in the process of familiarizing ourselves with its behavior in experimental setups. We will soon expand these experiments into the construction of a single bush actuator. When we are satisfied with its design, we will scale it to build second-level twigs.