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Design of Lightweight Lead Screw Actuators for Wearable Robotic ApplicationsJournal of Mechanical DesignKevin W. Hollander Thomas G. SugarA wearable robot is a controlled and actuated device that is in direct contact with its user. As such, the implied requirements of this device are that it must be portable, lightweight, and most importantly safe. To achieve these goals, The design of the standard lead screw does not normally perform well in any of these categories. The typical lead screw has low pitch angles and large radii, thereby yielding low mechanical efficiencies and heavy weight. However, using the design procedure outlined in this text, both efficiency and weight are improved; thus yielding a lead screw system with performances that rival human muscle. The result of an example problem reveals a feasible lead screw design that has a power to weight ratio of 277 W/kg, approaching that of the dc motor driving it, at 312 W/kg, as well as a mechanical efficiency of 0.74, and a maximum strength to weight ratio of 11.3 kN/kg 。 1 Introduction One in five persons in the United States live with some form of disability, with 61% of those suffering from either a sensory or physical disability.As an example, within the elderly population,8% to 19% are affected by gait disorders . Many disabled individuals could benefit from some form of robotic intervention. A wearable robot is a computer controlled and actuated device that is in direct contact with its user. The purpose of such a device is the performance/strength enhancement of the wearer. It can be used in training, in therapy, or simply as a device to assist in functional daily living. The implication of the term “wearable” isthat the robot must be portable, lightweight, and most importantly safe. In contrast, a factory floor robot is none of these things, so the simple adaptation of existing technology is not possible. The standard approach to wearable robot design suffers from three major limitations;1 Low battery power density；2 motors with low “strength to weight” ratios；3 weight and safety of a mechanical transmission system. The goal of this work is to review the design process of a lead screw actuator; the result of which will demonstrate significant improvements over the limitations described in item number 3, i.e., the weight and safety of the mechanical transmission system.2 BackgroundInterest in the area of wearable robotics has grown over the last decade. The recent surge of interest can be attributed to advancements in electronic miniaturization, microprocessor capabilities, and wireless technology proliferation. The feasibility of a portable computer controlled strength enhancing device is closer to reality However, aside from the availability of portable computation platforms, issues of the physical mechanism must still be addressed. The main issues in any wearable robot development are power, weight, and safety. How much power is available to do mechanical work? How much additional weight does the robotic device add to the person? And, how can this power be transferred and still maintain safety? The safe interaction between the wearer and theactuated robot has to be the primary concern in a wearable robot design.The purpose of a wearable robotic system is to offset the effort or energy of the operator by some amount of energy from a storage device, i.e., battery, fuel cell, and air tank. The sharing of the work load between the operator and the robot is heavily influenced by actuator efficiencies and the overall system weight. The additional weight that the robot adds to the user, in many cases, can increase the total amount of work required to accomplish a given task. This means that the robot not only has to augment the operators abilities, but must also compensate for its own additional weight.2.1 Actuator Comparisons. Human skeletal muscle is the “gold” standard by which many robotic actuators are compared. Known for their good “power to weight” ratios and excellent force production capabilities, skeletal muscle performance is what most actuator designers would like to match. In order to match the performance capabilities of skeletal muscle, it is important to know some of its measures. Unfortunately, common throughout biological literature is a wide variation of measured muscle properties. Although reported values have a wide variance, these values can still give a sense of scale in which biological materials behave. Data tabulated and estimated from several sources were used to describe the attributes of human muscle performance, and the result of which can be seen in Table 1.Table1：Actuator comparison: Compares various actuator types by mechanical efficiency, power to weight ratio, “corrected”power to weight ratio, and strength to weight ratio Measuresallows the direct comparisons to be made based upon utilization of available energy. However, both of these parameters need to be examined in the development of a wearable robotic actuator. Consider that if all actuators were to operate at 100% efficiency, then the entire group could be compared directly by their respective power to weight ratios. However, if only the power stated in the power to weight ratio were supplied to each actuator, then because of their respective efficiency, only a fraction of that power would be yielded as output. Therefore, to appropriately compare the above described actuators, their corrected power to weight（c） ratios must be computed (1)where is the mechanical efficiency and Pwt is the original power to weight ratio. The results of this calculation for various kinds of actuators can be seen in Table 1.Values in Table 1 were obtained either by referenced literature or estimations based upon that literature. The values for the dc motor are for the Maxon RE40 motor. The values for the + gearbox combination were also found in the Maxon 2004 catalog. values from an electric Series Elastic Actuator were used to estimate these parameters. However, a similiarly sized lead screw system will likely have a better strength to weight ratio, due to its ability to carry higher loads and its nut is of lower weight. For the McKibben style air muscles, a variety of literature was found describing its relevant measures.Immediately evident in this comparison is that the corrected power to weight, cP, values of the dc motor, the air muscle and human skeletal muscle are all similarly matched. However, once additional hardware is added to the dc motor, its performance decreases significantly. If one could create a mechanical transmission system that did not significantly alter the weight of the dc motor based actuator, then performances very near that of human skeletal muscle could be achieved.3 Lead Screw Design。Seen above, the performance of a typical lead screw system is limited when compared to other wearable robotic actuator concepts. The primary reason for its low performance is poor mechanical efficiency. The coefficient of friction in a standard lead screw system is approximately =0.36., metal on metal, better results are possible if lubrication is used.In contrast, the typical ball screw system has very good mechanical efficiency. The rolling contact of the ball bearings keeps the frictional effects on this system to an absolute minimum. However, even with its improved efficiencies, the cP value for the ball screw actuator is still well below that of skeletal muscle, due directly to the considerable weight of the ball screw system. To improve the cP performance of a ball screw, a significantreduction of weight must be achieved.Journal of Mechanical Design Fig. 1 Lead screw geometry; as drawn, pitch p and lead lare equivalent in a single helix screw The basic mathematics surrounding the design of a lead screw can also apply to a ball screw system. The primary difference between these two mechanical transmissions is their coefficient of friction. In the following section, an exploration of the design parameters that influence weight and mechanical efficiency of a lead screw will be considered and thus improvements to its ccan be made.3.1 Lead Screw Geometry.Shown in Fig. 1 is the basic geometry of a common lead screw. The key parameter of a lead screw is the lead, l, which is dependent on screw radius, r, and lead angle. The lead, l, is the amount of displacement achieved for each revolution of the screw. A high precision screw has a very short or fine lead. The right triangle in Fig. 1 shows the unwrapped geometry of a single revolution of a screw. The lead angle , represents the incline or slope of the screw thread. The base of the triangle is the circumference of the screw shaft, the right leg of the triangle is its lead, and the hypotenuse representsthe path length of the helical thread. Also seen on the right triangle are the forces present on a screw that is lifting a load. The force of the load is shown as Fw, the force resulting from the torque on the screw is F, the normal reaction force on the thread of the screw isN, and the frictional force is N. From this diagram, the following equation for a lifting torque can be derived (2) 3.2 Alpha Versus R.Considering, again, the geometry of a lead screw in Fig. 1, it can be shown that leadl, is described both by screw radiusr, and lead angle. The relationship between these variables is given in (3) （4）The meaning of Eq（4）is that both r, screw radius, and, lead angle, are necessary to create a screw lead, l. This means that there exists a continuous relationship between r and . Although this continuous relationship exists, most screw systems are designed with very small lead angles. A review of the preferred ACME screw sizes reveal that although the individual diameters vary, the lead angles are all less than 3.From Eq（4）.it is shown that for any screw lead desired, a variety of radii could be used. The significance of this is that as screw radius, r, shrinks, the weight of the screw shrinks by a factor.r2 Thus, to compensate for small screw radii, a larger value of lead angle , must be considered.Fig. 2 Mechanical efficiency of lead screw systems: Shaded part of the graph is the typical design region for the majority of lead screws. is small, radius is large, weight is large, and efficiencies are lower. Designs in the unshaded region of the graph, where is large, implies smaller radii, lower weight, and higher efficiencies. 3.3 Efficiency Versus Alpha. For a wearable robot design, not only is the weight of a lead screw actuator an important issue, but the efficiency of an actuator is also key. As mentioned before, a decrease in screw radius can achieve significant reductions in actuator weight. However, while the screw radius is reduced, the lead angle, must be increased to maintain a constant lead. When looking at Eq(2). it is seen that the torque required to lift a load, Fw, is dependent upon both lead angle, as well as the coefficient of friction。Relating the efficiency of a screw to both lead angle and coefficient of friction, Figure 2 shows the impact on both coefficient of friction, and lead angle, on the efficiency of a lead screw system （5）Each line in Fig. 2 is based upon a different value of the coefficient of friction. Several common engineering materials are given as examples to give the reader a sense of what effect different materials or coatings could have on the efficiency of a lead screw system. This figure shows that as the lead angle increases, so does the mechanical efficiency; or at least until a peak value is reached.Ideally, it would be advantageous to pick the angle, based upon maximum efficiency. A lead screw system operating at peak efficiency minimizes the input torque requirements to lift the load Fw. The angle at which peak efficiency occurs can be determined by taking the derivative of efficiency with respect to angle, the result of which can be seen in (6)Although a high lead angle can lead to a high efficiency, it can also lead to a system that is “back-drivable”. A back-driveable system is one in which the load, Fw, can cause a rotation of the screw without the assistance of applied torque, thus allowing the load, Fw, to self-lower. A back-driveable lead screw is a bad idea for a car jack, but is desirable in a wearable robot. For the lead angles in which back-drive will occur (7)Lead angle and coefficient of friction are all that influence this condition, regardless of how high the load force becomes. Fora very low coefficient of friction system, such as a ball screw,back-drive is an inevitable consequence.4 Practical ConsiderationsIdeally, as shown in the previous text, it would be desirable to reduce our screw radius, r, to an almost microscopic scale. However, this is not a practical solution, neither from a design nor manufacturing perspective. Although small screw diameters and high lead angles are desired from the perspective of weight and efficiency, they may not allow the designer to meet the strength demands of the physical system. Issues, such as axial yielding,compression buckling, and mechanism bind, need to be considered as well. Consider that a single ultrathin screw may be lightweight, although it may not be strong enough to carry the load required by the system. A single or several screws can be used, but must be sized large enough to handle the load placed upon it. As a note,there is no weight advantage to using several small screws to carry a large load, as the computation for both weight and stress are driven by a cross-sectional area of the screw. However, using several small screws to carry the load can allow the continued use of high lead angles and thus operate with high efficiencies, even in the presence of high loads. By pushing the limits of raw material properties of the lead screw, high axial loading can be achieved. This approach works better for a tensional system than it does for a compression bearing system. When considering the compressive loading of a long slender screw, Euler buckling must be addressed . Similar to that of the McKibben actuators or even human muscles, a lead screw actuator could be designed to bear a tensional load only, thus eliminating the consideration of buckling altogether. Creating a tension-only actuation system in a wearable robot does not necessarily mean that an antagonistic pair is required. In fact, for an assistance robot, a disabled person may only have muscle weakness in a single actuated direction and, therefore, a single tensional actuator would be all that is required to aid that person.。For those designers who would push the limits of the screw radius and thus lead angle to beyond that of maximum efficiency, the presence of friction limits just how far the angle can be inclined. The physical interpretation of this is that the system willbind or lock. This can be seen by evaluating Eq.（2）. An evaluation of the denominator in Eq.（2）. yields the following relation。 (8) In addition to the practical considerations listed here, there exists many other issues that could be detailed. Examples of which may include torsional stiffness/yielding or even heat dissipation. Each of these factors are important and worthy of consideration, however, the purpose of this exercise is to demonstrate an alternativeto the typical approaches of designing or selecting screw systems. The benefits of this alternative approach are directly applicable to the design issues of a wearable robotic system.5 Example Problem To demonstrate a crude design exercise, consider the peak ankle joint torque during gait of an able-bodied or normal individual that weighs 80 kg and walks at 0.8 Hz stepping frequency. The peak ankle torque during gait is approximately 100 Nm. This peak occurs at roughly 45% of the gait cycle, A gait cycle is defined by the heel strike of a foot to the next heel strike of the same foot. Toe off is the point in which the weight of the individual has transferred to the opposite leg and the initiation of swing begins. The conclusion of the swing phase of gait places the foot back into a heel strike position again and then the next gait cycle can begin.As an example, let us consider building a lead screw actuator for ankle gait assistance. For our problem, let us assume the levelTable 2： Example problem actuator comparison: Compares lead screw designs I and II to human muscle in terms of mechanical efficiency, power to weight ratio, corrected power to weight ratio and strength to weight ratio, measures of assistance to be at 30% and that the actuator acts with a 12 cm moment arm to the ankle joint. These values can be changed but, based upon personal experience, are reasonable in their scale. Using these values and parameters available for a chosen Maxon motor, the RE40, a range of lead lengths for this example solution has been determined; the range of possible screw leads areExample Problem Results. Two lead screw designs were generated to solve this problem. The first design, lead screw I, is a design solved for maximum efficiency. Assuming a lead of 2 mm and a =0.05, yields an efficiency of 0.9 for the screw at =43.5 and a radius of 0.34 mm. With such a small radius, multiple screws are needed to hold the load. Even so, estimates for the actuator power to weight are 280 W/kg. Power to weight has been determined by dividing the peak power required in our example by the weight of the motor and estimated transmission system. From our previous work, the weight of the accessory components was scaled proportionally to the reduced weight of the screw and nut.The second design, lead screw II, uses dimensions available from a commercial vendor. The screw is estimated to have an =13.6 and an efficiency of 0.82. Even with these larger dimensions, the actuators power to weight ratio of 277 W/kg =0.74 is expected. The results of this example problem have been tabulated for the purpose of comparison. Table 2 shows the numerical results of both example lead screw designs. These values are compared to the previous values tabulated for a dc motor alone, and the estimated values for human skeletal muscle. The strength to weight properties calculated for these examples is based upon the peak force required by our example.6 DiscussionIn the analysis of the maximum efficiency solution, lead screw design I, it was shown that a single small radii screw will not always handle the loads required of it. However, a bundle of screws operating in parallel can perform that task with the same high efficiency. Although a 0.34 mm radius screw would not be easily manufactured using typical techniques, it is possible that this kind of approach i.e., use multiple screws to maintain high efficiency could be useful for a MEMs scaled device. One could imagine a compact “force pack” built up from many high efficiency small diameter screws. Without going to the extremes in efficiency for a particular screw design, it was shown that for lead screw design II, a feasible solution exists for our example problem of ankle gait. Corrected power to weight values were obtained that are very close to those discussed for human muscle. Using a similar approach, a ball screw mechanism could benefit in performance, as well. The general approach to creating back-driveable, low weight, and high efficiency screw system can make a dc motor-based actuator a competitive solution f