A Novel Variable Transmission with Digital Hydraulics

Transcription

1 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Congress Center Hamburg Sept 28 - Oct 2, Hamburg, Germany A Novel Variable ransmission with Digital Hydraulics Zhenyu Gan, Katelyn Fry, R. Brent Gillespie, Member, IEEE, and C. David Remy, Member, IEEE Abstract his paper presents a novel variable transmission system that is based on the concept of digital hydraulics. In the proposed system, sets of rolling-diaphragm cylinders are mounted via different effective lever arms to an input and output joint. A variable subset of these cylinders is connected via three-way two-position on/off valves to a common hydraulic manifold. his introduces a controllable constraint on the hydraulic flow and creates a programmable hydraulic transmission. With three single-acting cylinders, we could realize 37 different transmission ratios. We investigated the nonholonomic flow constraint analytically, in simulation, and with an experimental prototype. Using water as fluid, we show that a very stiff transmission (124.2 Nm/rad) can be achieved within the range of ±6. heoretical transmission ratios are tracked with R-squared values of more than and backlash is smaller than 1.4%. Furthermore, we show the applicability of the proposed transmission in the simulation of a body-powered knee-ankle exoskeleton. I. I NRODUCION Actuation remains one of the great challenges in robotic systems. Particularly for systems that require large torques at slow velocities, the currently preferred solution of electric motors is not always ideal [1], [2]. While electric motors are well-suited for computerized control, they can only produce low torques and thus require large gearboxes or other transmission systems. he drawbacks of using multi-stage gearboxes include a large reflected inertia, high impedance, backlash, and overall low efficiency. Additionally, these added components lead to a low torque-to-weight ratio. [3]. hese drawbacks are particularly undesirable for embodied portable robotic devices such as rehabilitation robots or exoskeletons [4], [5]. Hydraulic systems can overcome some of these drawbacks as they have a much higher torque density and thus do not require additional transmissions. Yet, traditional hydraulic actuators introduce substantial mechanical losses due to the large friction forces created by the O-ring seals. Additionally, the closed-loop control of hydraulic actuators typically relies on the use of spool valves. hese valves generate a large pressure drop at the valve, which leads to substantial power losses. Especially when systems are operated close to a steady state, where large pressure drops across a very small orifice require higher supply pressure and input power. his typically require system components over-sizing, and external cooling systems may be needed.[6]. he authors are with the Robotics and Motion Laboratory (RAMlab), Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI (ganzheny@umich.edu, katelyf@umich.edu, brentg@umich.edu, cdremy@umich.edu) Funding for this project was provided by NIH (GRAN: 1-R01-EB Wearable embots to Induce Recovery of Function ) /15/$ IEEE Fig. 1. he experimental prototype for this project shown in (a) consists of two pendulums as input and output. Different combinations of the active cylinders can be used to achieve a large number of different transmission ratios. he CAD model of this setup is shown in (b). o overcome the problem of fluid power losses in spool valves, researchers have proposed digital hydraulic technology [7], [8], [9]. Rather than using continuously varying servo-valves, digital hydraulic technology typically relies on multiple simple on/off-components that are connected in parallel. Different flow-rates and active cylinder areas are achieved via a binary coding of differently sized components [9]. An example is a Digital Fluid Control Unit (DFCU) that mimics the function of a spool valve via an array of simple solenoid valves [10]. hese properties make digital hydraulics an interesting solution for a variable hydraulic transmission system. Whitney et al. [11] have recently presented a highly efficient hydraulic transmission with a fixed transmission ratio. heir solution is based on pairs of rolling diaphragmsealed cylinders that are acting in a antagonistic fashion. In contrast to traditional cylinders, the pistons of these cylinders are sealed with a diaphragm made of nitrile rubber. his con- 5838

2 struction greatly reduces mechanical friction in the seals. he authors have demonstrated a fixed transmission between two pairs of cylinders that are connected directly with each other and were pre-pressurized. his master-slave configuration allows a nearly loss-less transmission over large distances and is virtually free from hysteresis, backlash, and reflected inertia. Such a transmission allows one to locate a heavy torque source away from the actuated joint, thereby enabling lightweight systems with low inertia. However, since the transmission ratio is fixed to a constant value, external power and control on the drive side are needed to create a desired motion. In the present paper, we demonstrate how these two recent developments can be combined into a novel concept that allows variable transmissions between two robotic joints. o this end, we combine the actuation concept of the passive fluid transmission (as presented in [11]) with a digital valve bank (as proposed in [9]) to create a lightweight and efficient variable transmission. We connect a number of rolling diaphragm-sealed cylinders to the input and output joints of our system. he cylinders have different lever arms and thus create different relations between joint velocities and hydraulic flow-rates, and consequently between joint torques and hydraulic pressures. A bank of 3/2-way solenoid valves allows a variable connection between these cylinders, which realizes a large number of different transmission ratios between the input and output joints. For a suitable choice of lever arms a smooth set of discrete transmission ratios (both positive and negative) can be achieved. he resulting transmission is lightweight, has very low impedance, and can be modulated in its compliance by a choice of the hydraulic fluid and hose specifications. his novel variable hydraulic transmission can be useful in a wide range of robotic applications in which power must be routed between joints, or from a motorized drive to a joint. Our work is inspired in particular by the idea of bodypowered exoskeletons, in which such variable transmissions are used to route power from healthy limbs to impaired limbs or to prosthetic devices. he variable transmission allows computer control of the transmission and thus enables a large variety of possible motion trajectories. Since the system is completely passive, it is inherently safe and provides direct force-feedback to the user. An example of such a system is shown in figure 6. Since the constraints are enforced on velocity-level and not with respect to positions, the variable transmission generates a variable, nonholonomic constraint. In other words, it allows the tracking of arbitrary output trajectories as long as there is some motion on the input joint. In the remainder of this paper, we will present the underlying mathematical theory of creating velocity constraints via digital hydraulics. We demonstrate the use of our approach with a system composed of 6 single acting cylinders and present experimental data for multiple transmission ratios. Additionally, we use a detailed simulation of the hydraulic system to show the usage of the proposed system in the problem of tracking a healthy ankle trajectory by using the Fig. 2. hree cylinders are connected to the input and output joint, respectively. In this particular example (which shows our experimental setup) all cylinders have the same cross section A i = 248 mm 2, but different lever arms l 1 = 25.4 mm, l 2 = 47.6 mm, and l 3 = 69.9 mm. Via a set of valves, the cylinders are either connected to a common closed manifold (shown: cylinders 1, 2, 4, and 6 with pressure p M ) or to an open reservoir (shown: cylinders 3 and 5 with pressure p R ). his is expressed by a binary selection vector s = [ ]. For this valve setting, the transmission ratio of the system is = α/ β = knee motion as power source. II. HEORY A. Creating Kinematic Constraints he hydraulic system that creates the variable transmission consists of n single acting cylinders with individual cross sections A i. he cylinders are connected to a rigid body system with m joints with joint angles θ j. his is shown in figure 2, for a system with two joints (θ = [α β] ) and 6 cylinders. he displacement x i of each piston is a kinematic function of the joint angles x = f (θ). Consequently, the piston velocities ẋ depend in a linear fashion on the joint angle velocity θ: ẋ = f θ. θ Assuming that the fluid is incompressible, the flow rate of each cylinder is given by Q i = ẋ i A i. Defining the matrix A as a diagonal matrix of all n cylinder cross sections, we can compute a flow-rate Jacobian J R n m that yields the hydraulic flow rate as a function of joint motion according to: Q = J θ, with J = A f θ. (1) his Jacobian determines the functionality of the variable transmission. It can be designed for a specific application by choosing appropriate cross-sections, lever arms, and the overall kinematic configuration of the system. he net flow of all cylinders Q = n i=1 Q i can be expressed in vector notation as: Q = 1 J θ = j θ, (2) with 1 being an n-dimensional vector of ones used in the dot product to sum over the columns of J. he m-dimensional vector j = J 1 is comprised of the sums over the columns of J. o create a variable transmission between joints, each cylinder is either connected to a common manifold or to an open reservoir via a set of n three-way valves. Let the binary vector s represent this set of valve settings. If a valve is connected to the common manifold, it is represented in s by a 1, otherwise by a 0. he sum of all flows going into the common manifold must be equal to zero, which 5839

3 enforces a kinematic constraint on velocity level. We express this constraint as the dot product: s Q = s J θ = c θ = 0, (3) with the m-dimensional constraint vector c = J s. Since this flow is zero, the flow into the reservoir is equal to the total flow: Q R = Q = j θ. (4) For a simple system with only two joints θ = [α β], the constraint defined by eq. (3) can be interpreted as a variable transmission. α β = c 2 =, (5) c 1 in which c 1 and c 2 are altered by changing the valve settings in s. B. Forces and Pressure he transpose of the flow-rate Jacobian J, maps a vector of cylinder pressure values p to joint torques τ : τ = J (p 1p A ), (6) where p A is the ambient pressure of the system. Let p M denote the pressure in the common manifold and p R the pressure in the reservoir. Assuming a loss-less system, with no pressure-drops along the hydraulic lines, we can rewrite eq. (6) as: τ = J [sp M + (1 s) p R 1p A ] = (7) = J s (p M p R ) + J 1 (p R p A ) = = c (p M p R ) + j (p R p A ). he total power P that is produced by this system is given by: P = τ θ = (8) = c θ (pm p R ) + j θ (pr p A ) = = Q R (p R p A ), as c θ = 0 per eq. (3) and Q R = j θ per eq (4). his is the power that the system performs by pushing fluid into the pressurized reservoir. Since we are interested in a pure transmission between joints, ideally this power would be equal to 0. We can achieve this by either setting the reservoir pressure p R equal to p A, or by sizing the cylinders and the kinematic structure to yield a Jacobian with j = 0. Note: Nonlinearities in the kinematics x = f (θ) can lead to a Jacobian J (θ) that changes as a function of the joint angles θ. In this case, the requirement j = 0 can only be fulfilled approximately. Under these assumptions, a simple system with two joints that experiences torques τ = [τ α τ β ], enables a transmission of torques according to τ α = c 1. (9) τ β c 2 he pressure p M in the manifold that transmits these torques is given by p M = p R + τ α = p R + τ β (10) c 1 c 2 o avoid negative pressure in the rolling diaphragm cylinders in the presence of torques τ α and τ β with varying signs, a suitable high value for p R must be chosen to ensure p M p A 0. Such a pressurization of the system has potentially additional benefits. For example, it leads to a stiffer transmission and reduces backlash [11]. Yet, as shown before, it is only possible for systems in which j = 0. C. Design of the Flow-Rate Jacobian Within this setup, different transmission ratios are achieved by changing the valve setting in s. With n valves and cylinders, 2 n settings can be achieved that are expressed by 2 n different binary vectors of s. Some of these values might be redundant. For example, any configuration of s that does not connect cylinders on the output side to the common manifold will result in a transmission = 0. A suitable choice of lever arms and cylinder cross sections allows tuning these transmission ratios to create a smooth variation over a desired range. It is notable that our formulation also allows for bi-articular cylinders that are connected to both the input and output joint. Such cylinders would create multiple entries in the flow-rate Jacobian J. III. HARDWARE IMPLEMENAION o evaluate the theory of the proposed variable transmission system, we implemented a simple experimental setup with two joints (m = 2) and a total of six rolling diaphragm cylinders (n = 6). he joints are simple pin joints with a simple lever attached to them. hree cylinders were connected to each lever (Fig. 1). he cylinders used in this setup were Small Bore Diaphragm Cylinders (Control Air Inc., Amherst, NH) small bore cylinders with an effective pressure area of A i = 248 mm 2 and a stroke length of approximately 18 mm. hese cylinders were assigned different lever arm lengths (l 1 = 25.4 mm, l 2 = 47.6 mm, l 3 = 69.9 mm, Fig. 2). In order to achieve both positive and negative transmission ratios, the cylinders were oriented such that one cylinder per lever arm was acting in opposition to the two other cylinders. his setup leads to a linearization of the Jacobian around the vertical position of: [ ] J = cm 3 /rad For this system j = [ ] 0; that is, the total flow is not zero and we should chose p R = p A. Since the lever arms are identical on both the input and the output joint, some of the 2 6 valve settings resulted in redundant transmission ratios. Still, this simple system allows 37 different transmission ratios (including transmission ratios of zero and infinity), spanning the range to (Fig. 3). Each cylinder was connected to a 3/2 solenoid directional valve (ASCO, General Service Solenoid Valves, 8320G130). When the solenoid is energized (s i = 1), the valve connects the corresponding cylinder to the common hydraulic manifold with pressure p M otherwise (s i = 0) to the reservoir with pressure p A. We used water as working fluid and 5840

4 Fig. 3. Shown are the sorted transmission ratios that arise from the 26 different valve settings of the example system shown in figure 2, a total of 37 different transmission ratios spanning from to A finer resolution or larger range can potentially be obtained by optimizing the lever arms or adding more cylinders to the system. Fig. 4. We evaluated the stiffness of the proposed hydraulic transmission by fixing the input shaft of the experimental prototype and varying the torque τα of the output side while recording the output angle α. We formulated a fairly linear stiffness law which approximates our system as a linear spring. he linearized stiffness of this system with a transmission ratio of -1 is Nm/rad. conducted experiments in which the fluid in the reservoir was not pressurized (pr = pa ). Since in this configuration, eq. (10) can potentially yield negative pressures for varying loads, we limited ourselves to a unidirectional transmission of forces. o this end, we applied constant positive loads τα to the output joint. he range of motion for both input and output joints was kinematically limited to ±15 deg. Joint angles were measured with incremental encoders (Avago echnologies, HEDR-55L2-BY09) with a resolution of 0.1 deg. IV. E XPERIMENAL E VALUAION A. ransmission Stiffness In a first experiment, we quantified the transmission stiffness of the system. We chose a transmission ratio of -1 (s = [ ] ), blocked the input side (β 0), and gradually increased the load torque τα via weights and a pulley system on the output lever. he load was varied from -2.0 Nm to 2.0 Nm in increments of 0.2 Nm, and the deflection of the output side α was measured (Fig. 4). A linear regression analysis evalutes the stiffness of this transmission to be τα / α = Nm/rad. B. ransmission Accuracy In a second set of experiments, we manually moved the input lever in an approximately sinusoidal fashion, and recorded the joint-angle trajectories for both sides. By connecting a subset of the cylinders to the common manifold and Fig. 5. he input/output behavior of the experimental prototype was recorded using three different static transmission ratios of = 1, = 2, and = he output side was subject of a constant load of 2.03 Nm and the input was driven manually to approximate a sinusoid of 0.25 Hz (shown in (a)) and 1 Hz (shown in (b)) for the three transmission cases. he dotted red dotted lines show the theoretically expected transmission values. the remainder to the reservoir, we tested three different trans mission ratios of = 1 s = [ ], = +2 s = [ ], and = 3.75 s = [ ]. he input/output behavior of the transmission is shown in figure 5 for slow (0.25 Hz) and fast (1 Hz) inputs. Experiments were conducted against a constant load torque of 2.03 Nm. When compared to a theoretically perfect transmission, in which α = β (dotted lines in Fig. 5), only a small error can be observed. he R2 values for the three cases are ( = 1), ( = +2) and ( = 3.75). he hysteresis band of the system is also very small, particularly for small transmission ratios. From these experiments, the maximal backlash values for the three transmission ratios are rad ( = 1), 5841

5 Fig. 7. A simple P-controller with a velocity feed-forward component was used to track a desired ankle angle α. A desired transmission ratio des is obtained from the ratio of desired ankle joint velocity and knee joint velocity. Valve settings s are selected from a look-up table to match des as close as possible. Fig. 6. A possible application for the proposed variable hydraulic transmission are body-powered exoskeletons. In this particular example, a hydraulic transmission spans the knee and ankle and allows to route power from a healthy knee joint to an impaired ankle joint; for example, to prevent drop-foot after stroke. he two sets of cylinders are connected selectively by a valve bank during a stride to realize an appropriately changing transmission ratio that creates a desired ankle motion rad ( = +2) and rad ( = 3.75). V. S IMULAION OF A K NEE -A NKLE E XOSKELEON In preparation to running dynamical tests of the variable transmission (which would require a motorization of the experimental prototype and computer control of the valves, both of which are not implemented in the current prototype), we conducted a simulation study of the system to evaluate a potential control strategy and estimate dynamic behavior. o this end, we implemented a model of the experimental prototype in MALAB SimHydraulics (the Mathworks, Natick, MA). his framework simulates the complete hydraulic flow, including valve switching. In the model, we used water as hydraulic fluid. Flow viscosity was set to 0.48 cst, and the relative amount of trapped air to %. he valve maximum opening is 1.2 mm. All cylinders were modeled as single acting cylinders with a piston area of 248 mm2. As example, we chose a trajectory tracking task for the example application of a body-powered knee-ankle exoskeleton (Fig. 6). his system harvests power at the user s healthy knee joint and routes it in a nonholonomic fashion to an impaired ankle joint; either on the same limb or on the contra-lateral side. By continuously varying the hydraulic transmission ratio between knee and ankle, the system is able to track an arbitrary ankle trajectory; that is, as long as the knee joint is moving and providing power. Potential applications of such a system include the prevention of drop-foot in stroke patients. For the underlying hydraulic transmission system, we assumed a system that is equivalent to our experimental prototype. However, rather than using six single acting cylinders (three on each joint), an equivalent system could potentially be implemented using three double acting cylinders; for example, with two cylinders at the knee and one cylinder at the ankle (as shown in Fig. 6). We considered the knee motion of an average healthy person to constitute the input trajectory β (t), and a healthy ankle motion as the desired output trajectory α (t). As controller, we propose a simple proportional controller with a feed-forward component, that computes a desired transmission ratio des (t) according to: des = α + P (α α) β (11) From a look-up table, a set of valve settings s that best approximates des is selected at a frequency of 100 Hz (we assume the stride to last 1.2 s). A block diagram of controller and plant is shown in figure 7. We chose the proportional gain P to be 10 1/s. he simulation of a complete stride cycle, the choice of valve settings, and the resulting transmission ratios are shown in figure 8. Even with a fairly simple system with only three singleacting cylinders per joint, a close tracking of a desired motion is possible. On average, the motion created by the transmission deviates only by rad from the reference. Larger tracking errors are only accumulated, when the velocity of the input (i.e., the knee joint) is close to zero. hese cases would require large (possibly infinite) transmission ratios which cannot be realized with any pure transmission system. In this particular example this happens around 16 %, 40 %, and 77 % of the stride cycle. VI. D ISCUSSION AND C ONCLUSION In this paper, we presented a novel variable transmission that is based on the concept of digital hydraulics. his transmission is implemented via a set of single-acting rolling diaphragm cylinders that are connected via a valve bank to a common hydraulic manifold. he flow constraint in this manifold creates a variable nonholonomic constraint between input and output joints. he experimental results from initial tests of the transmission prototype are highly encouraging. Due to the high stiffness of the hydraulic fluid, the system has a high torque density, a good transmission accuracy, and very little backlash. he system is very stiff, has a good transmission accuracy, and exhibits very little backlash. Because of the rolling diaphragm cylinders, the transmission also has a low impedance and very little friction. In combination, these properties mean that the transmission is very efficient, which makes it particularly suitable for embodied robots such as prosthetic applications or exoskeleton systems. Even with a naive proportional controller that did not take into account the complexity of the nonholonomic control and planning problem, precise trajectory tracking was realized. he fairly simple system with only six cylinders was capable 5842

6 configuration. Reducing the switching frequency can cause error to accumulate in the system. Choosing an appropriate feedback control that steers away position errors that might accumulate can potentially help overcome the need for such a high switching frequency and increase the limited resolution with respect to the transmission ratio. In the presented prototype, the transmission consisted of six identical cylinders with identical lever arms on both the input and output side. As a result, the system showed some redundancy in the achievable transmission ratios. hat is, multiple valve configurations resulted in identical transmission ratios. By optimizing the lever arms (or the choice of cylinder cross section), the flow-rate Jacobian J can be optimized to yield a more extended and/or smoother set of transmission values. Such an optimization of J will be part of our future work, which also aims at the implementation of dynamic control in actual hardware. his includes extensions of the current prototype, and work in geometric control that would enable the realization of stable trajectory tracking with a more reasonable valve-switching frequency. Fig. 8. Shown is the simulation of a full walking stride of a body-powered knee-ankle-exoskeleton (Fig. 6). he knee motion (shown as joint angle in (a)) constitutes the input β (t) to the system. By creating a time-varying nonholonomic constraint via the proposed variable hydraulic transmission, this creates a driven ankle trajectory (shown in (b), black) which is tracking a normative, healthy motion (shown in (b), red). he selected valve settings s (shown in (c)) and transmission ratios (shown in (d)) are altered at a frequency of 100 Hz. of providing a close tracking of a complex desired motion trajectory. For comparison, a hydraulic transmission with a fixed transmission ratio (as it presented in [11]) needs already two cylinders per joint and is not much less complex. he largest deviations shown in figure 8 occured when the velocity of the input, β, was close to zero. In these cases, the necessary transmission ratios become very large. his is a structural problem of any passive transmission, as the required transmission ratio reaches infinity in the limit case of zero input velocity. While in theory, the achievable transmission ratio can easily be extended beyond the range from to 4.26, the scaling of the associated forces makes such large transmission ratios not particularly applicable. For such cases, allowing the temporary storage of energy (for example, in an additional spring loaded cylinder) could become a viable option. he controller in the simulation currently uses an update rate of 100 Hz, which is near the upper limit of switching frequency of the valves used in the current hardware REFERENCES [1] W.. ownsend and J. K. Salisbury, Mechanical design for wholearm manipulation, in Robots and Biological Systems: owards a New Bionics?, pp , Springer, [2] I. W. Hunter, J. M. Hollerbach, and J. Ballantyne, A comparative analysis of actuator technologies for robotics, Robotics Review, vol. 2, [3] S. Seok, A. Wang, D. Otten, and S. Kim, Actuator design for high force proprioceptive control in fast legged locomotion, in Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, pp , IEEE, [4] W. K. Durfee and A. Rivard, Design and simulation of a pneumatic, stored-energy, hybrid orthosis for gait restoration, Journal of biomechanical engineering, vol. 127, no. 6, pp , [5] R. B. Gillespie, J. L. Contreras-Vidal, P. A. Shewokis, M. K. O Malley, J. D. Brown, H. Agashe, R. Gentili, and A. Davis, oward improved sensorimotor integration and learning using upper-limb prosthetic devices, in Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE, pp , IEEE, [6] D. N. Johnston, A switched inertance device for efficient control of pressure and flow, in ASME 2009 Dynamic Systems and Control Conference, pp , American Society of Mechanical Engineers, [7] J. Mattila and. Virvalo, Energy-efficient motion control of a hydraulic manipulator, in Robotics and Automation, Proceedings. ICRA 00. IEEE International Conference on, vol. 3, pp , IEEE, [8] M. Linjama, A. Laamanen, and M. Vilenius, Is it time for digital hydraulics?, in In: Koskinen, K & Vilenius, M.(eds.) he Eight Scandinavian International Conference on Fluid Power, Proceedings of the Conference, May 7-9, 2003, ampere, Finland, SICFP 03, [9] M. Linjama, Digital fluid power: State of the art, in 12th Scandinavian International Conference on Fluid Power, ampere, Finland, May, pp , [10] M. LINJAMA and M. VILENIUS, Energy-efficient motion control of a digital hydraulic joint actuator, in Proceedings of the JFPS International Symposium on Fluid Power, vol. 2005, pp ,, [11] J. P. Whitney, M. F. Glisson, E. L. Brockmeyer, and J. K. Hodgins, A low-friction passive fluid transmission and fluid-tendon soft actuator, in Intelligent Robots and Systems (IROS 2014), 2014 IEEE/RSJ International Conference on, pp , IEEE,