ME-599 Design Project Final Report Brian Trease
Statically Balanced Compliant Four-Bar Mechanism Ercan Dede
May 11, 2004
Abstract
We designed and built a prototype of a torsion-spring-based statically-based 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% at this time) 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.
Statically Balanced Systems Review
Statically balanced systems 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. The most significant takeaway “tool” for the design of statically balanced systems is this concept of controlling the system’s potential energy through design such that it is constant.
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. Through the physical embodiment, these springs are typically used to balance a payload potential energy function that varies non-linearly. As was shown in our lecture, the concept of zero-free-length springs must be discarded when designing compliant statically balanced systems.
For more on the concept of zero free length springs please refer to Just Herder’s dissertation [1] and website (http://www.wbmt.tudelft.nl/mms/wilmer/herder.htm). Since most of Herder’s work requires a linear spring approach, we must start again from the beginning to incorporate compliant mechanisms.
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 after in the past, 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. It is the goal of this project to achieve similar results to Herder, but using a yet unproven approach – compliance via torsion springs.
Figure 1: Single Degree of Freedom Gravity Balancer (from Herder)
The specific goal of this project was to design a statically-balanced single-degree-of-freedom compliant system. This simple system can be thought of as a medical or orthotic assistance device, reducing the muscle force required to move an arm about its elbow.
The primary goal was to be able to move a 0.5 kg (1.1 lb) mass through a 90 degree range of motion (±45 degrees from horizontal) at a 200 mm (7.87 in) moment arm. The desired balancing error was to be <1% or <.01 mg of operating force at the mass. Figure 2 below is a rigid link version of this system developed by Herder [2]. Thus, we are merely trying to find another solution to the same problem Herder examined, but we seek a solution that will have the additional benefits that compliant mechanisms offer. (e.g., zero-backlash, no wear, friction-free, repeatability), see Dede [3].
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 (i.e., the green curve) is obtain, which means that the system has 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, though non-zero.]
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 and final method (See Figure 4) was selected by virtue of its relative simplicity.
Figure 4: Schematic Representation of 4-Bar Mechanism
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 very 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
These four sets of variables are shown in Figure 5 and Figure 7. However, these concepts are much easier to understand by viewing the MS-PowerPoint animations in the file that accompanies this report. In both figures, the left-hand graph is joint displacement versus mass-arm angle, and the right-hand graph is joint potential energy versus mass-arm angle. View the right-hand side plots as the “square” of the left-hand side plots (multiplied by stiffness constants).
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.
Figure 5: 4-Bar Displacement and Energy Curves
(Zero System Shift & Joint Preload, Equal Joint Stiffnesses)
Figure 6: 4-Bar Mechanism Illustrating Practical Embodiment of System Shift
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 (i.e., over-stress) 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.
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. In the future, it is recommended to have a program that adds robustness to these equations, thus allowing the possibility of running an optimization routine during the design process.
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)
Figure 9: Cross-Section of “Open Cross” Revolute Joint
The constraints in step 2.b. (above) are that none of the joints be overstressed (i.e. all rotations stay within the calculated range of motion). 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.
The “open-cross” revolute joint used (See Figure 9) was developed by Brian Trease and Audrey Plinta, a modification of the design developed by Sridhar Kota, Yong-Mo Moon, and Brian Trease [5]. The formulas for stiffness and range of motion are below in Table 1. Please contact us for further information, such as off-axis (cross-axis) stiffness equations.
Table 1: Analytic Stiffness TableTorsional [N-mm/rad] Mx/qx / 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*sysL2Q/[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; sys~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 10: Fabricated Configuration /
Figure 11: Zero Degree Configuration
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. This figure shows the maximum required angles of joints 3 and 4 to be 37º and 17º, respectively.
Figure 14: Potential Energy Analysis of Balanced 4-Bar Mechanism Including Payload
Design Model
Prior to manufacturing the assembly a solid model was created in SDRC I-DEAS Version9. 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 a band saw. The total weight of the mechanism is 0.246 kg with the outboard 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.
A prototype was fabricated and presented. The device and its details are shown in Figure 16 though Figure 18.
Figure 15: I-DEAS Solid Model of Compliant 4-Bar Mechanism
Figure 16. Fabricated Prototype of Compliant 4-Bar Mechanism
Figure 17. Compliant Joint (#4) /
Figure 18. Compliant Joint (#3)
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 recommendation for future work is to manufacture a more refined prototype. This prototype should be designed such that there is greater robustness and less error introduced in to the system from manufacturing. Stiffer joint connections, improved construction (e.g., aluminum links), increased dimensional tolerances, and superior joint material (e.g., titanium) should be utilized. In addition, once this second generation prototype is manufactured, it would be worthwhile to quantify the balancing error in order to compare final results with the analysis above. Pending successful completion of the above, it is proposed to enter this design in to the 2004 ASME Student Mechanism Design Competition.
The second recommendation, perhaps to be executed in conjunction with the first, is to increase the fidelity of the analytical model. Specifically, this requires accounting for non-linear stiffness effects of the “open cross” revolute joints in the analysis. Additionally, exact material properties (i.e., Elastic Modulus, Shear Modulus, etc.) for the beams in the compliant revolute joints should be determined from either the material vendor or manufacturer.