2004 ASME Student Mechanism Design Competition

Statically-balanced Compliant Four-bar Mechanism for Gravity Compensation

Team Members: Eric Dede and Brian Trease The University of Michigan

Graduate Category Faculty Sponsor: Sridhar Kota

Contact: Brian Trease

Dept. of Mechanical Engineering 2350 Hayward / Room 2231

Ann Arbor, MI 48109 trease@asme.org

Abstract We designed and built a prototype of a torsion-spring-based statically-balanced gravity compensator. This was achieved using novel “open-cross” compliant joints in a four-bar mechanism. The mechanism supports a mass at the end of a moment arm, effectively canceling out the force of gravity. This means that a human can move the mass as if it were weightless. Our balancing is nearly constant (max. error is 3%) over a rotation range of ±45º. To our knowledge, this is the first such torsion-spring based device, and our results serve as the motivation for continuing research. The implications of this project are significant in that they point to the development of innovative compliant systems capable of static-balancing, which may help widen research possibilities in this field. Statically-Balanced Systems

Statically-balanced systems, first studied by Just Herder [1], are considered energy free in that there is full energy exchange between storage elements in an ideal sense. Excluding non-conservative effects such as friction, statically-balanced systems have constant potential energy throughout the entire range of motion. As a result, these mechanisms have no preferred position, which implies that they require zero effort to operate in a quasi-static sense.

A key concept in traditional statically-balanced systems is the idea of a zero-free-length

spring where both force and deflection are proportional to length [1]. A clever trick of trigonometry allows for these springs to greatly simplify the problem. However, the concept of must be discarded when designing compliant statically-balanced systems. We must start again from the beginning: the concept of controlling the system’s potential energy through design such that it is constant. Applications Most applications for statically-balanced mechanisms are for systems that balance a gravity force and a spring force over a broad range of motion. (See Figure 1) These are “mass compensators” that can be used in robotics to reduce actuator forces. Similarly, such a system could aid workers on assembly lines or loading docks by “removing” the weight of a payload. This goal of mass balance has been sought before, but solved with bulky counterweight systems with their own set of problems. Finally, there are also medical applications, such as orthotic devices, that can reduce muscle requirements for partially-disabled patients. Design Criteria

Herder first designed a simple spring-balancer mechanism, depicted both schematically and in final form below. His design parameters are also listed. Our broad goal was to achieve results similar to Herder’s using a yet unproven approach – compliance via torsion springs.

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g

Figure 1: Single Degree of Freedom Gravity

This simple system of Figure 1 can be thought of a

device, reducing the muscle force required to move an arm

The primary goal was to be able to move a 0.5 kg (of motion (±45 degrees from horizontal) at a 200 mm (7.87error was to be <1% or <.01 mg of operating force at the mversion of this system developed by Herder [1]. Thus, we solution to the same problem Herder examined, but we seeadditional benefits that compliant mechanisms offer. (e.g.repeatability.)

Statically-Balanced Compliant Four-Bar Mechanism

“zero-free-length” sprin

m

f = mg

Balancer (from Herder)

s a medical or orthotic assistance about its elbow.

1.1 lb) mass through a 90 degree range in) moment arm. The balancing ass. Figure 2 below is a rigid link are merely trying to find another k a solution that will have the , zero-backlash, no wear, friction-free,

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Figure 2: Rigid Link Statically-Balanced 1-DOF System (from Herder)

Understanding the Energy Balance

The key to solving this design problem is to understand the energy within the system. The blue curve in Figure 3 below represents the potential energy of the mass when resolved about the pivot point. Notice that it varies nonlinearly as the cosine of the pivot angle. The pink curve represents the energy of the 4-bar system that we wished to generate. When these two lines are added, a straight, horizontal line (the green curve) is obtained, indicating a constant potential throughout its range of motion. [Note that it is acceptable if the pink link is shifted up or down the vertical axis; the resulting system energy will still be constant.]

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Potential Energy Fucntions vs. Angle

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

-50 -40 -30 -20 -10 0 10 20 30 40 50

Angle (deg)

Pote

ntia

l Ene

rgy

(N-m

) Mass PE4-Bar PESystem PE

Figure 3: Energy Curves for Statically-Balanced 4-Bar Mechanism

Design Alternative & Options

Three primary design alternatives were examined as possible solutions to the problem. The first was to use a non-linear torsion bar to generate a variable stiffness joint. This could possibly be accomplished by utilizing a protrusion along the span of a rod to vary the effective torsion length. The second concept was to incorporate a leaf spring deflected against a cam to vary the effective length of the beam, thus achieving a variable stiffness joint. The third and final concept was to use a 4-bar mechanism with various combinations of compliant joints to achieve an overall mechanism energy function that varies non-linearly. The third method (See Figure 4) was selected by virtue of its relative simplicity.

Figure 4: Schematic Representation of 4-Bar Mechanism

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Design Method

The pink, dotted line in Figure 3 above shows the stiffness required of our system, which is part of a sinusoid and thus non-linear / non-quadratic. Torsion joints by themselves have linear deflections and thus quadratic potential energy curves. In order to go from quadratic potential energy to nonlinear, there are several design variables that can be adjusted. Design Variables

• System shift • Joint Preloads • Joint Stiffness • Link geometry

System shift refers to the angle at which the mass-supporting arm attaches to the top link

of the four-bar mechanism. (See Figure 6) System shift moves the entire set of both curves left or right along the mass angle axes. Joint preload is load added to each joint by twisting it in place, effectively moving a particular joint’s displacement curve left or right along the x-axis, as well as its energy curve. Joint stiffness simply affects the curvature of the quadratic potential energy curve.

Figure 5 and Figure 7 show joint rotations and energies for parallelogram four-bar mechanisms; all of the angular relationships are linear with respect to the mass angle.

Figure 5 shows a system with no shift or joint preload, and all stiffnesses equal. The resultant potential energy is then just a parabola, shown in orange. Figure 7 shows a system with shift (black arrow), joint preloads (colored arrows), and varying stiffnesses (different curvatures). The sum of these curves is not shown; although it is known that it will be quadratic since the sum of any number of quadratic curves can only be another quadratic curve.

Joint Displacement PE

Input Crank Angle

Input Crank Angle

Figure 5: 4-Bar Displacement and Energy Curves

(Zero System Shift & Joint Preload, Equal Joint Stiffnesses)

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System Shift Angle α

Figure 6: 4-Bar Mechanism Illustrating Practical Embodiment of System Shift

PE Joint Displacement

Input Crank Angle

Input Crank Angle

Figure 7: 4-Bar Displacement and Energy Curves (Non-Zero System Shift & Joint Preload, Unequal Joint Stiffnesses)

In order to get nonlinear potential energy, we must change the quadratic shape of the

parabolas in Figure 7. This can only be accomplished by changing the linear angular relationships, also shown in Figure 7. These curves quickly become nonlinear once we change the geometry of the mechanism away from that of a parallelogram. Preferred Kinematics

Because we are looking at joint compliance to balance the system energy, we must be careful not to over-rotate the joints and break them. Thus, it is desirable to have compliance only in the joints that undergo the least amount of rotation. Therefore, the top-ground joint will be a pivot, since it must go through ±45 degree rotation. By Grashof’s law, we know that if the input link and the coupler link are the shortest, then the last two joints (i.e., 3 and 4) will undergo the least amounts of rotation. Such a layout is shown in Figure 8, and is the basic configuration that was utilized in design iterations.

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Method of Kinematics While it is easy to generate the closed-form analytic equations describing relative joint

rotations for any particular 4-bar mechanism, it is difficult to do this with robustness in general. This is because of the two sets of solutions that can arise due to inverted mechanism arrangements.

Joint 2 Joint 1

Joint 4

F=mg Joint 3

Figure 8: Basic 4-Bar Topology Minimizing Rotation at Joints 3 and 4

In lieu of the above, the multi-body dynamics software ADAMS was used to obtain the

non-linear displacements, which were then exported into Excel to allow adjustment of the other design variables. The basic design loop implemented is as follows:

1. Kinematics (from ADAMS) 2. a. System shift, preload, and stiffness (in Excel)

b. Constraints (in Excel)

The constraints in Step 2.b. are that none of the joints be overstressed. The range of

motion is actually coupled to the joint stiffness, requiring an iterative process of adjusting values and checking that both calculations are satisfied.

Thickness (t)

Width (w)

Figure 9: Cross-Section of “Open Cross” Revolute Joint

The “open-cross” revolute joint used (See Figure 9) was developed by one of the authors

and a lab-mate, a modification of a design developed by the same author and his advisor. The formulas for stiffness and range of motion are below in Table 1.

gap

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Table 1: Analytic Stiffness Table Torsional [N-mm/rad] Mx/θx 24 EI1(w+g)2/L3 + 8 GK/L Bending [N/mm] Fy/dy, Fz/dz 48 E(I1+I2)/L3

Axial [N/mm] Fx/dx 2 AE/L Range Of Motion [±rad] 0.577*σysL2Q/[2.25(EQt)2(w+g)2 + 3(KGL)2]1/2

Note: I1=1/12*wt3; I2=1/12*tw3; A=4wt; K = wt3/16 [16/3 – 3.36 t/w (1-t4/ (12w4))]; Q = w2t2/[3w+1.8t] E~Young’s Modulus; G~Shear Modulus; σys~Yield Strength

Results

After several design iterations, the potential energy of the system was balanced over ±45º within a 3% error (occurring only at the limits of motion). This solution was achieved using only two compliant joints and zero preload in both. The system only required a system shift, thus eliminating the need for joint preload and reducing manufacturing complexity. A 79 degree system shift was used. Figure 10 shows the mechanism in its fabricated configuration (no moment yet in any of the joints). Figure 11 thru Figure 13 show the mechanism in its zero, plus 45, and minus 45 degree configurations. The deflection of the joints in the last 3 figures provides the spring force that balances the variable moment from the mass.

Figure 11: Zero Degree Configuration Figure 10: Fabricated Configuration

79°

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Figure 12: +45 Degree Configuration Figure 13: -45 Degree Configuration

The final potential energy analysis of the system, including mass, is shown in Figure 14.

The mass curve (Mass PE) also includes the masses of all the mechanism links, to ensure accurate balancing. Another point to note is that the resulting maximum required angles of joints 3 and 4 were minimized at 37º and 17º, respectively.

-1000

-500

0

500

1000

1500

2000

-45 -30 -15 0 15 30 45Mass Angle (deg)

Pote

ntia

l Ene

rgy

(N-m

m)

Joint 3Joint 4Total 4-Bar PEMass PEmass + 4B + links

Figure 14: Potential Energy Analysis of Balanced 4-Bar Mechanism Including Payload

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Design Model

Prior to manufacturing the assembly a solid model was created in SDRC I-DEAS Version 9. All parts were manufactured from 0.100” thick acrylic except the open-cross beams which were manufactured from 0.125” thick polypropylene. The acrylic was laser cut while the polypropylene was machined using standard shop practice. The total weight of the mechanism is 0.246 kg with the outboard (i.e., moving) portion weighing 0.136 kg. As mentioned above, link mass was accounted for in the final balancing of the system, and only increased balancing error by ~1.5%. Figure 15 provides an image of the final design model.

Open Cross Beams: machined Polypropylene

End Caps: laser cut Acrylic

Links (yellow/green): laser cut Acrylic

Link 4 - Fixed to Ground

Link 3 – 2X

Link 2

Link 1 –Ground Pivot

Links attachedusing 4X 0.086-56

fasteners

Spacer (2X)

P=0.5 kg

Figure 15: I-DEAS Solid Model of Compliant 4-Bar Mechanism

The fabricated device and its details are shown in Figure 16 though Figure 19.

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Figure 16. Fabricated Prototype of Compliant 4-Bar Mechanism

Figure 17. Compliant Joint (#4) Figure 18. Compliant Joint (#3)

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Figure 19. Close-up of rigid links and pin-joints.

Conclusions and Recommendations

The design above clearly illustrates that a particular solution to this problem is possible. This design ultimately is a “proof-of-concept” in that general compliant mechanism solutions to statically-balanced systems are a reality. The first step for future work is building a more refined prototype for the design competition. Stiffer joint connections, improved construction (e.g., aluminum links), increased dimensional tolerances, and superior joint material (e.g., titanium) will be utilized, providing less error and more robustness. This prototype will allow us to quantify the balancing error and compare it with our analyses. The second step, to be executed in conjunction with the first, is to increase the fidelity of the analytical model. This requires accounting for non-linear stiffness of the “open cross” revolute joints. Optimization

We suggest coding a closed-form solution of the kinematics to implement within an optimization routine. This would allow the designer to minimize the geometry while reducing balancing errors to less than one percent. Thus far, we’ve only used 7 out of 12 available design variables, showing considerable opportunity to find more accurate and compact designs.

References

[1] Herder, J. (2001) Energy-free Systems. Theory, conception, and design of statically-balanced spring mechanisms. PhD-thesis. Delft University of Technology, Delft, The Netherlands.

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