Exploiting Natural Dynamics to Reduce Energy Consumption by

The International Journal of Robotics Research http://ijr.sagepub.com

Exploiting Natural Dynamics to Reduce Energy Consumption by Controlling the Compliance of Soft Actuators Bram Vanderborght, Björn Verrelst, Ronald Van Ham, Michaël Van Damme, Dirk Lefeber, Bruno Meira Y Duran and Pieter Beyl The International Journal of Robotics Research 2006; 25; 343 DOI: 10.1177/0278364906064566 The online version of this article can be found at: http://ijr.sagepub.com/cgi/content/abstract/25/4/343

Published by: http://www.sagepublications.com

On behalf of:

Multimedia Archives

Additional services and information for The International Journal of Robotics Research can be found at: Email Alerts: http://ijr.sagepub.com/cgi/alerts Subscriptions: http://ijr.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav

Downloaded from http://ijr.sagepub.com at PENNSYLVANIA STATE UNIV on February 5, 2008 © 2006 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution.

Bram Vanderborght Björn Verrelst Ronald Van Ham Michaël Van Damme Dirk Lefeber Bruno Meira Y Duran Pieter Beyl Vrije Universiteit Brussel Department of Mechanical Engineering Pleinlaan 2, 1050 Brussels, Belgium [email protected] http://lucy.vub.ac.be

Abstract Exploiting natural dynamics for bipedal locomotion, or passive walking, is gaining interest because of its energy efficiency. However, the natural trajectories of a passive walker are fixed during the design, thus limiting its mobility. A possible solution to this problem is creating a “semi-passive walker” equipped with actuators with adaptable compliance, which allows the natural dynamics to be changed according to the situation. This paper proposes a compliance controller, a strategy for continuously changing the compliance in such a way as to adapt the natural motion of the system to a desired trajectory. This opens up the possibility of following a range of different trajectories with a relatively low energy consumption. The idea is to fit the controllable actuator compliance to the “natural” compliance of the desired trajectory, and combine that with trajectory tracking control. This strategy was implemented and tested on a 1-DOF pendulum setup actuated by an antagonistic pair of pleated pneumatic artificial muscles. Both simulations and measurements show that the proposed strategy for choosing actuator compliance can significantly reduce the amount of control activity and energy consumption without harming tracking precision.

KEY WORDS—compliance control, stiffness control, soft actuator, compliant actuator, exploiting natural dynamics

1. Introduction Increasingly, robots are moving from factories where they act in a human-free environment towards places where they have to share their working environment with humans e.g., in domestic, entertainment, assistance, rehabilitation, or medical The International Journal of Robotics Research Vol. 25, No. 4, April 2006, pp. 343-358 DOI: 10.1177/0278364906064566 ©2006 SAGE Publications

Exploiting Natural Dynamics to Reduce Energy Consumption by Controlling the Compliance of Soft Actuators

applications. This means that absolute accuracy is not that stringent anymore, while other aspects like energy reduction and safety become more and more important. A promising approach is to put the compliance in the hardware by using compliant actuators, also called soft actuators, which makes the joint intrinsically safer. In this paper the natural dynamics of the system will be exploited by controlling the compliance of the actuator. 1.1. Compliance for Bipedal Locomotion Bipeds can be divided into two main categories: on one side there are the completely actuated robots which do not use natural or passive dynamics at all, like Asimo (Hirose et al. 2001), Qrio, HRP-2 (Yokoi et al. 2003) and on the other side there are the “passive walkers” who do not need actuation at all to walk down a sloped surface or only use a limited amount of actuation when walking over level ground, just enough to overcome friction. Examples are the Cornell biped, the Delft biped Denise and MIT robot Toddler (Collins et al. 2005). The main advantage of “passive walkers” is that they are highly energy efficient and energy consumption is an important issue for autonomous bipeds. Unfortunately they are of little practical use. They have difficulties starting, cannot change their speed and cannot stop, contrary to a completely actuated robot. Probably the optimal will be somewhere in between those two opposites. As in human locomotion this can be achieved by continuously adapting the natural dynamics. To do this we use actuators with controllable compliance just as humans do. 1.2. Compliant Actuators Electrical drives make the joints stiff because they need a gearbox to deliver enough torque at low rotation speeds. 343


344

THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / April 2006

Compliant or soft actuators cope with this by placing elastic elements in the actuation. An important contribution in the research towards compliant actuators has been given by Pratt with the development of the “series elastic actuator” (Pratt and Williamson 1995). It consists of a motor drive in series with a spring and has been successfully implemented in the two legged robot “Spring Flamingo” (Pratt and Pratt 1998). In this setup the stiffness is fixed. Nowadays research is focused on actuators with adaptable compliance. Takanishi developed the two-legged walker WL-14 (1998) (Yamagushi et al. 1998), where a complex non-linear spring mechanism makes predefined changes in stiffness possible. Hurst et al. of the Robotics Institute at Carnegie Mellon University developed the “Actuator with Mechanically Adjustable Series Compliance” (AMASC) (Hurst et al. 2004). It has fiberglass springs with a high energy storage capacity. The mechanism has two motors, one for changing the position and one for controlling the stiffness. The electro-mechanical Variable Stiffness Actuation (VSA) (Bicchi and Tonietti 2004) motor developed by Bicchi and Tonietti of the University of Pisa is designed for safe and fast physical human/robot interaction. A timing transmission belt connects nonlinearly the main shaft to an antagonistic pair of actuator pulleys connected to positioncontrolled backdrivable DC motors. The belt is tensioned by springs. Concordant angular variations control displacements of the main shaft, while the opposite variations of the two DC motors generate stiffness variations. The Biologically Inspired Joint Stiffness Control (by Migliore et al. 2005 of the Georgia Institute of Technology, USA) can be described as two antagonistically coupled Series Elastic Actuators, where the springs are made non-linear. At Northwestern University the Moment arm Adjustment for Remote Induction Of Net Effective Torque (MARIONET) is being developed by Sulzer et al. (2005). This rotational joint uses cables and a transmission to vary the moment arm such that the compliance and equilibrium position is controllable. What is special is that this device doesn’t use an elastic element, the system moves against a conservative force field created by a tensioner. The mechanisms with variable compliance mentioned above are relatively heavy and large to be used in mobile robots. A more elegant way to implement a variable compliance is to use pneumatic artificial muscles. These muscles can only pull, thus in order to have a bidirectionally working revolute joint, one has to couple two muscles antagonistically. Using artificial muscles, the applied pressures determine position and stiffness. Research at the Vrije Universiteit Brussel focuses on the Pleated Pneumatic Artificial Muscle (PPAM, Daerden and Lefeber 2001). To study the promising advantages of artificial muscles as adaptable compliant actuators for walking robots, the biped Lucy (Figure 1) was built (Verrelst et al. in press). It weighs about 30 kg and is 150 cm tall. It uses 12 muscles to actuate 6 degrees of freedom, so Lucy is able to walk in the sagittal plane only. A guiding mechanism prevents the robot

Fig. 1. Photograph of Lucy.

from falling sidewards. The goal of this project is to create a lightweight biped which is able to walk in a dynamical stable way while exploiting the adaptable passive behavior of the pleated pneumatic artificial muscles in order to reduce energy consumption and control actions. Some videos of the walking robot can be seen at the website: http://lucy.vub.ac.be. In this paper a strategy is proposed to change the compliance in order to reduce control actions and energy consumption. The idea is to fit the controllable compliance of the muscles to the natural compliance of the desired trajectory. The combination of changing the natural dynamics of the system by the compliance controller and a joint trajectory tracking controller is implemented on a pendulum setup. The paper is organized as follows. In Section 2, we describe the pleated pneumatic artificial muscles and the torque and compliance characteristics of an antagonistic muscle setup. Section 3 describes the combination of the joint trajectory tracking controller and the compliance controller. Section 4 illustrates the concepts with simulation and practical experiments showing that the compliance controller reduces valve actions and energy consumption. Finally, Section 5 provides conclusions.


Vanderborght et al. / Exploiting Natural Dynamics to Reduce Energy Consumption

2. Pleated Pneumatic Artificial Muscles 2.1. Force Characteristics A pneumatic artificial muscle is essentially a membrane that expands radially and contracts axially when inflated, while generating high pulling forces along the longitudinal axis. Different designs have been developed. The best known is the so called McKibben muscle (Schulte 1961). This muscle contains a rubber tube which expands when inflated, while a surrounding netting transfers tension. Hysteresis, mainly due to dry friction between the netting and the rubber tube, makes control of such a device rather complicated. Typical of this type of muscle is a threshold level of pressure before any action can take place. The main goal of the Pleated Pneumatic Artificial Muscle (PPAM) design (Daerden and Lefeber 2001) was to avoid friction, thus making control easier by avoiding the threshold and reducing hysteresis. This was achieved by arranging the membrane into radially laid out folds that can unfurl free of radial stress when inflated. Tension is transferred by longitudinal fibers that are positioned at the bottom of each crease. A photograph of different contractions of the PPAM is given in Figure 2. If we omit the influence of the very small elasticity of the high tensile strength material used for the fibers, the generated force is given by: l F = pl 2 f , (1) R where p is the applied gauge pressure, l the muscle’s full length, R its unloaded radius and the contraction. The dimensionless force function f depends only on contraction and geometry. The graph in Figure 3 gives the generated force for different pressures of a muscle with initial length 110 mm and unloaded diameter 25 mm. Forces up to 5000 N can be generated with gauge pressure of only 3 bar while the device weighs about 100 g. At low contraction, forces are extremely high causing excessive material loading, and for large contraction the generated forces drop very low. So, in practice, contraction is limited between 5 and 35%. The generated forces are much higher at lower pressure levels compared with the McKibben and Festo muscles. Their working principle is the deformation of a reinforced rubber tube instead of volume increase by unfolding the pleats of the PPAM. For example a McKibben muscle with a diameter of 6 mm and a length when fully stretched of 210 mm can generate maximum 200 N at 3.5 bar. A Festo muscle with an internal diameter of 10 mm and a nominal length of 100 mm can generate up to 630 N at 3.5 bar. The PPAM also has a higher contraction range.

and lever mechanism—were designed in such a way the muscle’s non-linear force-length characteristic is transformed to a more linear torque-angle characteristic. Figure 4 shows the straightforward connecting principle. The dimensions of the muscles and the lever arm can be chosen in order to meet the specified joint characteristic, not only torque level but also range of motion. The torque characteristics of the 1 DOF pendulum used for the test setup are shown in Figure 5. Here one torque is generated by the agonist muscle T1 and the other by the antagonist muscle T2 . Both torques are given for different pressure values. Taking into account eq. (1) and if r1 and r2 define the lever arm of the agonist and antagonist muscle respectively, the joint torque is given by following expression: T = T1 − T2

= =

p1 l12 r1 f1 − p2 l22 r2 f2 p1 t1 (θ) − p2 t2 (θ )

(2)

with p1 and p2 the applied gauge pressures in the agonist and antagonist muscles respectively having lengths l1 and l2 . The dimensionless force functions of both muscles are given by f1 and f2 . The functions t1 and t2 , in eq. (2), are determined by the choices made during the design and depend on the angle θ . Thus joint position is influenced by weighted differences in gauge pressures of both muscles. 2.3. Compliance Characteristics of an Antagonistic Muscle Setup The PPAM has two sources of compliance: gas compressibility and the dropping force to contraction characteristic (Daerden and Lefeber 2001). The latter effect is typical for pneumatic artificial muscles while the first is similar to standard pneumatic cylinders. Joint stiffness, the inverse of compliance, for the considered revolute joint can be obtained by the angular derivative of the torque characteristic in eq. (2): K

= =

dT1 dT2 dT = − dθ dθ dθ dp1 dt1 dt2 dp2 t1 + p 1 − t 2 − p2 dθ dθ dθ dθ

(3)

The terms dpi /dθ represent the share in stiffness of changing pressure with contraction, which is determined by the action of the valves controlling the joint and by the thermodynamical processes. If polytropic compression/expansion with closed valves is assumed, then the pressure changes inside the muscle will be a function of volume changes:

2.2. Torque Characteristics of an Antagonistic Muscle Setup Pneumatic artificial muscles can only pull. In order to have a bidirectionally working revolute joint one has to couple two muscles antagonistically. The muscle connections—pull rods

345

Pi Vin = Pio Vino

(4)

Pi = Patm + pi

(5)

with:


346


Fig. 2. Photograph of 3 contraction levels of the PPAM.

7000 1 bar 2 bar 3 bar 4 bar

6000

Force (N)

5000

4000

3000

2000

1000

0

0

10

20 Contraction ε (%)

30

Fig. 3. Generated forces (N).


40


Fig. 4. CAD drawing of the basic frame with muscles and connection plates.

80 1 bar 2 bar 3 bar 4 bar

70

Torque (Nm)

60 50 40 30 20 10 Antagonistic Muscle 0

-20

Agonistic Muscle

0 Joint Angle (°)

20

Fig. 5. Agonistic and antagonistic muscle torque at different pressures.


347

348


leading to:

3.1. Joint Trajectory Tracking Controller Vino dVi dpi = −n Patm + pio dθ Vin+1 dθ

(6)

with Pi , Vi the absolute pressure and volume of muscle i, Pio the absolute initial pressure, Vio the initial volume when the valves of muscle i were closed and pi , pio the gauge pressure and initial gauge pressure; n is the polytropic index and Patm the atmospheric pressure. Taking the torque characteristics as an example the following reasoning can be made for muscles with closed valves. An increase of the angle θ will result in an increase of the torque generated by the agonistic muscle (see Figure 5) while its volume will decrease. Thus dt1 /dθ > 0 and dV1 /dθ < 0. For the antagonistic muscle the actions will be opposite. Combining eqs. (3), (4) and (6) with this information gives: K = (k1 p1o + k2 p2o + katm Patm )

(7)

with: k1

=

k2

=

katm

=

V1no dV1 V1n dt1 | | + no | | n+1 dθ V1 dθ V1 n V2o dV2 V2n dt2 t2 n n+1 | + no | | | dθ V2 dθ V2 dt1 dt2 k1 + k 2 − | | − | | dθ dθ t1 n

>0 >0

The coefficients k1 , k2 , katm are a function of the joint angle and are determined by the joint and muscle geometry. From eq. (7) the conclusion is drawn that a passive spring element is created with an adaptable stiffness controlled by the weighted sum of both initial gauge pressures when closing the muscle. Since stiffness depends on a sum of gauge pressures while position is determined by differences in gauge pressure, the angular position can be controlled while setting stiffness.

3. Trajectory Control with Adaptable Compliance In this section the proposed strategy of combining the adaptable passive behavior with a trajectory tracking controller is illustrated. The control architecture consists of two parts: a joint trajectory tracking controller that calculates the necessary torque to track a desired trajectory and a compliance controller that reduces the energy consumption and the valve switching. An extended version of the considered joint trajectory tracking controller has successfully been used in the biped Lucy (Vanderborght et al. in press). For the biped the compliance parameters have so far been set to a constant value, but the final goal is to implement the compliance controller proposed in this paper in the biped.

This paragraph discusses the joint trajectory tracking controller, depicted in Figure 6. The tracking controller consists of a computed torque controller, a delta-p unit and a pressure bang-bang controller. The computed torque controller calculates the required joint torque based on the pendulum dynamics. The delta-p unit translates these calculated torques into desired pressure levels for the two muscles of the antagonistic set-up. Finally the bang-bang controller determines the necessary valve actions to set the correct pressures in the muscles. In the next sections these control scheme building blocks are discussed in more detail. 3.1.1. Computed Torque In the first module, the joint drive torques are calculated using the well known computed torque technique consisting of a feedforward part and a PID feedback loop (Slotine and Li 1991). This results in the following calculation: T˜ = Ce θ, θ˙ θ˙ + Ge θ + De θ θ¨˜ − Kp θ − θ˜ − Ki θ − θ˜ − Kd θ˙ − θ˙˜

(8)

The parameters De , Ce and Ge contain the estimated values of the inertia, centrifugal, coriolis and gravitational parameters. The feedback parameters Kp , Ki and Kd are manually tuned. ∼ means desired value. 3.1.2. Delta-p Unit The computed torque T˜ is then used as input for the delta-p control unit, which calculates the required pressure values to be set in the muscles. These two gauge pressures are generated as follows: p˜ s + p˜ ˆt1 (θ ) p˜ s − p˜ p˜ 2 = tˆ2 (θ) p˜ 1 =

(9a) (9b)

with p˜ s a parameter that is used to control the sum of pressures and consequently the joint stiffness and p˜ a parameter that controls the difference in pressure of the two muscles in order to control the generated torque. The functions tˆ1 (θ ) and tˆ2 (θ ) represent the torque characteristics of the antagonistic muscle setup and are calculated with estimated values of the muscle force functions and geometrical parameters. Expression (2) allows us to link the required torque to the required pressure values in the muscles: T˜ = p˜ 1 tˆ1 (θ) − p˜ 2 tˆ2 (θ ) = tˆ1 (θ) + tˆ2 (θ) p˜ (10) If the calculated pressure values p˜ 1 and p˜ 2 of eqs. (9) are set in the muscles, the generated torque depends only on p˜ and



349

Fig. 6. The applied control architecture.

is independent of the joint stiffness parameter p˜ s ; in this case the modelling would be perfect. This means that joint stiffness can be changed without affecting the joint angular position. Feeding back the joint angle θ and introducing the torque T˜ , eq. (10) is solved for the required p˜ : p˜ =

T˜ tˆ1 (θ ) + tˆ2 (θ )

(11)

The delta-p unit is actually a feedforward calculation from torque level to pressure level, using the kinematic model of the muscle actuation system and the muscle characteristics. The calculated p˜ controls the torque required to track the desired trajectory, while p˜ s sets the stiffness of the joint which will be discussed in Section 3.2. 3.1.3. Bang-bang Pressure Controller The weight of the valves controlling the muscles should be as low as possible. But since most pneumatic systems are designed for fixed automation purposes where weight is not an issue at all, most off-the-shelf proportional valves are far too heavy for this application. Thus a proper design of the pressure control is of great importance. In order to realize a lightweight, rapid and accurate pressure control, fast switching on-off valves are used. The pneumatic solenoid valve 821 2/2 NC made by Matrix weighs only 25 g. With their reported opening times of about 1 ms and flow rate of 180 Std.l/ min, they are among the fastest switching valves of that flow rate currently available. To pressurize and depressurize the muscles, which have a varying volume up to 400 ml, a number of these small onoff valves are placed in parallel. Obviously the more valves used the higher the electric power consumption, the price and also the weight will be. Simulations of the pressure control of a constant volume of 400 ml resulted in a choice of two inlet and four outlet valves. The different number of inlet and

outlet valves comes from the asymmetric pressure conditions between inlet and outlet, combined with the aim of creating equal muscle inflation and deflation times. To connect the 6 valves into one compact pressure regulating system two special collectors were designed. These collectors replace the original aluminum connector plates of the valves, resulting in a weight of the complete pressure regulating valve of about 150 g. Electronic speed-up circuits reduce opening and closing times of the valves. The pressure control is achieved with a bang-bang controller with various reaction levels depending on the difference between measured and desired pressure. If this difference is large, two inlet or four outlet valves, depending on the sign of the error, are opened. If this difference is small, only one valve is switched and when the error is within reasonable limits no action is taken, leaving the muscle in a closed state. The principle of this control scheme is depicted in Figure 7. More detailed information on the valve system can be found in the work of Van Ham et al. (2005). 3.2. Compliance Controller In this section a strategy to calculate an appropriate value of p˜ s is described. The idea is to fit the actuator compliance to the natural compliance of the desired trajectory. The natural stiffness of the desired trajectory K traj ectory , the inverse of the compliance, is calculated as the derivative of the torque T˜ necessary to track the desired trajectory with respect to the joint angle θ˜ . The torque T˜ is given by the inverse dynamics: K traj ectory =

˙˙ dT˜ d ¨ ˜ θ˜ θ˜ + Ge θ˜ = De θ˜ θ˜ + Ce θ, ˜ ˜ dθ dθ (12)

The actuator stiffness can be found by deriving eq. (10) with respect to θ˜ . K actuator =

dtˆ1 dtˆ2 dp˜2 dp˜1 tˆ1 + p˜1 − tˆ2 − p˜2 dθ˜ dθ˜ dθ˜ dθ˜


(13)

350


Fig. 7. Multi-level bang-bang control scheme with dead zone (a = –60, b = –25, c = –20, d = 20, e = 25, f = 60 mbar).

The required pressure slopes are a combination of eq. (6), valid for muscles with closed muscles, with eqs. (9): Vˆ1no dVˆ1 dp˜ 1 n Vˆ1no dVˆ1 = −p˜ s − (Patm + p) ˜ n n+1 tˆ1 Vˆ1n+1 dθ˜ dθ˜ Vˆ1 dθ˜ (14a) Vˆ2no dVˆ2 dp˜ 2 n Vˆ2no dVˆ2 = −p˜ s − (Patm − p) ˜ n n+1 tˆ2 Vˆ2n+1 dθ˜ dθ˜ Vˆ2 dθ˜

4. Simulations and Experimental Results In this section the performance of the proposed combination of the compliance controller and the trajectory tracking controller is discussed, both through simulations and experiments on a pendulum setup (Figure 14). First a methodology is described to calculate the energy consumption. In the case of pneumatics, this is not so obvious, since energy consumption depends on how air was pressurized by the power source.

(14b) When the eqs. (14) and (13) are combined and the actuator stiffness K actuator is set equal to the stiffness K traj ectory , p˜ soptimal can be derived. Note that the initial volume Vjo , when closing the muscle valves, is set equal to the actual volume Vj . p˜ soptimal =

K traj ectory − g1 p˜ − g2 g3

(15)

with

ntˆ2 d Vˆ2 d tˆ1 d tˆ2 ntˆ1 d Vˆ1 ˙ − + + g1 (θ˜ , θ˜ ) = − d θ˜ d θ˜ Vˆ1 d θ˜ Vˆ2 d θ˜ ntˆ1 d Vˆ1 ntˆ2 d Vˆ2 g2 (θ˜ , θ˙˜ ) = Patm − + Vˆ1 d θ˜ Vˆ2 d θ˜ n d Vˆ1 n d Vˆ2 1 d tˆ1 1 d tˆ2 ˙ g3 (θ˜ , θ˜ ) = − + + − tˆ1 d θ˜ tˆ2 d θ˜ Vˆ1 d θ˜ Vˆ2 d θ˜

The compliance controller is consequently a feedforward calculation. In the next section it is shown that this value p˜ soptimal reduces the energy consumption. The energy consumption depends also on the controller parameters, such as bang-bang pressure control reaction levels and feedback gains of the computed torque controller.

4.1. Calculation of Energy Consumption The energy consumption of compressed air depends not only on the air mass flows, but is related to the thermodynamic conditions of the compressed air supply source. One way to estimate the energy consumption is to calculate the exergy associated with the pneumatic air mass flow. Exergy is the maximum amount of energy, with respect to the surrounding environment, which can be transformed into useful work. For example, the air mass flow entering the muscle system comes from a compressed air reservoir at a certain temperature and pressure level. The surrounding environment is at atmospheric pressure. The exergy of the pneumatic power supply is then calculated as the minimal work needed to compress the atmospheric air to the pressure supply conditions. For a compressor, the minimal work needed to compress air from pressure level patm to p1 is done under isothermal conditions and can be calculated as follows (Rogers and Mayhew 1992): p1 ˙ air rTatm ln W˙ isotherm = m patm

(16)

hereby assuming the air to behave as a perfect gas. The symbol m ˙ air represents the total air mass flowing through the compressor, r is the dry air gas constant and Tatm is the temperature


Vanderborght et al. / Exploiting Natural Dynamics to Reduce Energy Consumption of the atmosphere expressed in degrees Kelvin. In our calculations, the absolute supply pressure level p1 was set at 7 bar, the atmospheric absolute pressure patm at 1 bar and the atmospheric temperature Tatm is 293 K. This is just one way to calculate the energy consumption. The absolute energy consumption is not the most important aspect, however. It is the reduction of the energy consumption that one can achieve when matching system and trajectory compliances which is the main focus. 4.2. Simulation Results 4.2.1. Simulator Most parts of the system are highly non-linear: force/torquecontraction of the muscles, the thermodynamic processes in the muscles, the mechanical load of the pendulum, so it is difficult to make a stability/convergence analysis. In order to evaluate the proposed control strategy while taking into account all these non-linearities, a hybrid simulator was created, which means that both the pneumatics and mechanics were put together in a dynamic simulation. In the model, the pressure buildup inside the muscle is represented by first order differential equations deduced from the first law of thermodynamics for an open system while assuming a perfect gas for the compressed air. The orifice valve flows were derived from the model presented by ISO635 (ISO 1989). The integration of these first order differential equations coupled with the mechanical differential equations of the pendulum and the muscle torque/contraction characteristics gives the torques. Doing so includes the muscle actuator characteristics which allows us to study the limitations in the motion of the pendulum. Detailed information on the simulation model can be found in Verrelst et al. 2005. To test the robustness, model parameter uncertainties are introduced in the simulation model, e.g. 5% on the mass and COG, 10% on the inertia and 5% on both torque functions of the antagonistic set-up. A time delay of 1 ms for closing and opening of the valves and a sampling time of 2 ms of the controllers were used.

351

the calculated psoptimal consumes less energy and that the joint error is smaller as well. The effect of choosing the optimal ps can also be seen in Figures 11–13. These figures show the valve actions taken by the bang-bang pressure controller. Note that in these figures closed valves are represented by a horizontal line depicted at 2.0, 2.6 and 1.5 bar pressure level respectively, while a small peak upwards represents one opened inlet valve, a small peak downwards one opened exhaust valve and the larger peaks represent two opened inlet or four opened outlet valves. The number of valve actions is significantly lower when the compliance setting is at the optimal value. On the right the desired and actual pressures are depicted. In the optimal case the desired pressure is nearly the natural pressure already present in the muscle. Only at certain instants a little energy input has to be provided to the system to increase the swing frequency. In the cases where the compliance setting is not optimal, significantly more valve actions are required. The imposed trajectory differs a lot from the natural movement of the pendulum causing a lot of valve switching and consequently energy dissipation. The actual motion, however, is the same because it is controlled by the joint trajectory tracking controller. 4.3. Experimental Results with a Single DOF Pendulum 4.3.1. Pendulum The complete set-up is shown in Figure 14. The structure is made of a high-grade aluminum alloy, AlSiMg1, and the design is exactly the same as the limbs of the robot Lucy, only the connection parameters of the pull rod and lever mechanism are different (see Section 2.2). The length of the pendulum is 45 cm, at the end there is a weight of 10 kg. The sensors are Agilent HEDM6540 incremental encoder for reading the joint position and two pressure sensors (Honeywell CPC100AFC), mounted inside each muscle. The controller is implemented on a PC and the National Instruments AT-MIO-16E-10 is used for data acquisition working at 500 Hz. 4.3.2. Results and Discussion

4.2.2. Results and Discussion To evaluate the proposed control architecture a sinusoidal trajectory is chosen with a linearly increasing frequency from 1.5 Hz to 2 Hz (chirp function). Figure 8 shows the desired trajectory and Figure 9 shows the desired angular velocity. Due to the increasing frequency the stiffness has to be adapted continuously. To show the effectiveness of the proposed compliance controller the results of an optimal ps as calculated with eq. (15) have been compared with the results obtained with some deviations on it (p˜ s = 1.5psoptimal and p˜ s = 0.5p˜ soptimal ). Figure 10 depicts these p˜ s values with changing frequency. The influence of the parameter p˜ s on the exergy consumption and mean angle error is given in Table 1. It is clear that

Figure 15 shows the total experimentally estimated exergy as a function of the parameter p˜ s for sinusoidal trajectories with different frequencies and amplitudes. It is clear there exists an optimal value for p˜ s and it is logical that for increasing frequencies the stiffness has to increase as well, which means that the optimal p˜ s also has to increase. Table 2 gives both the experimentally determined and calculated (by eq. (15)) optimal values of p˜ s for 3 different swing frequencies. The calculated optimal p˜ s gives a good approximation of the compliance parameter that is needed for minimal energy consumption. The difference between the calculated and experimentally determined p˜ s value can be decreased by a better parameter estimation.


352


6

2

4

1.9

1.8 0 1.7

Frequency (Hz)

Angle (°/s)

2

-2 1.6

-4

-6

0

5

10 Time (s)

15

1.5 20

Fig. 8. Desired angular trajectory of the pendulum.

Table 1. Exergy Consumption and Mean Angle Error for Different ps ps Exergy Consumption (J ) 1.50psoptimal 1.25psoptimal 1.00psoptimal 0.75psoptimal 0.50psoptimal

5.63 2.98 0.87 2.78 5.28

Mean Angle Error (◦ ) 0.122 0.113 0.064 0.079 0.090

Table 2. Experimental and Calculated Optimal Values of p˜ s Sine Wave Frequency (Hz)

experimental ps,optimal (Nm)

calculated ps,optimal (Nm)

1.5 1.75 2.0

12 19 30

12.6 21.4 31.7



800 600

Angular velocity (°/s)

400 200 0 -200 -400 -600 -800

0

5

10 Time (s)

15

20

Fig. 9. Desired angular velocity of the pendulum.

50 0.5*ps 40

ps

optimal

opimal

1.5*ps

optimal

ps (Nm)

30

20

10

0 1.5

1.6

1.7 1.8 Frequency (Hz)

1.9

Fig. 10. p˜ s as a function of the frequency of the sinusoidal trajectory.


2

353


2.5

2.5

2.4

2.4 Pressure (bar)

Valve action

354

2.3 2.2 2.1 2 1.9

Real pressure Desired pressure Valve action

2.3 2.2 2.1 2 1.9

0

5

10 Time (s)

15

20

4.5

4.75

5 Time (s)

5.25

5.5

Fig. 11. Valve action and detail of the pressure changes for p˜ s = psoptimal (sim).


2.9

Pressure (bar)

Valve action

2.9 2.8 2.7 2.6 2.5

2.8 2.7 2.6 2.5

0

5

10 Time (s)

15

20

4.5

4.75

5 Time (s)

5.25

5.5

Fig. 12. Valve action and detail of the pressure changes for p˜ s = 1.5psoptimal (sim).

2

2 Real pressure Desired pressure Valve action

Pressure (bar)

Valve action

1.9 1.8 1.6 1.4

1.8 1.7 1.6 1.5 1.4

0

5

10 Time (s)

15

20

1.3 4.5

4.75

5 Time (s)

Fig. 13. Valve action and detail of the pressure changes for p˜ s = 0.5psoptimal (sim).


5.25

5.5


Fig. 14. CAD drawing and photograph of the physical pendulum.

20

Exergy (J)

15

10

1.5 Hz, 5° 1.75 Hz, 5° 1.75 Hz, 10° 2 Hz, 5°

5

0

0

10

20 ps (Nm)

30

40

Fig. 15. Total output exergy vs stiffness parameter ps for different frequencies and amplitudes.


355


Angle (°)

356

6 4 2 0 -2 -4 -6

9

10

11 12 Time (s)

13

6 4 2 0 -2 -4 -6

9

10

11 12 Time (s)

13

6 4 2 0 -2 -4 -6

9

10

11 12 Time (s)

13

Fig. 16. Effect of closing all valves at t = 10 s when left: p˜ s = 12 Nm, middle: p˜ s = 19 Nm = psoptimal and right: p˜ s = 30 Nm.

Figure 15 also shows the total exergy as a function of the parameter p˜ s for two amplitudes of 5◦ and 10◦ at 1.75 Hz. The optimal stiffness stays nearly the same as we would expect for a pendulum, but energy consumption is higher for larger amplitudes. The actual passive trajectory of the pendulum deviates from a pure sine-wave. This deviation increases for larger amplitudes. Consequently, more valve switching is needed for larger movements, which causes a higher energy consumption. In another experiment the tracking controller tracks a sine wave trajectory of 1.75 Hz and after t = 10 s the controller is stopped by closing all the valves. This experiment has been repeated for different compliance settings. The pendulum with ps = 19 Nm (Figure 16, middle) (which is the optimal compliance for 1.75 Hz, see Table 2) will keep swinging with almost the same frequency as the imposed trajectory, after closing the valves. A higher ps compared with the optimal p˜ s means a stiffer joint, consequently the frequency increases; a lower ps makes the joint more compliant and thus the frequency decreases. The unforced amplitude of course decreases due to friction and leakage of the muscles. One can see that when p˜ s = 30 Nm (Figure 16, right), the pendulum starts oscillating after t = 10 s with a frequency of about 2 Hz, and when p˜ s = 12 Nm (Figure 16, left) the frequency is about 1.5 Hz. This is not surprising in view of the results in Table 2. To test the effectiveness of the proposed compliance controller the experiment with the chirp function trajectory going from 1.5 Hz to 2.0 Hz (see Section 4.2.2 for the simulations) is performed on the real pendulum. The results are given in Figures 17 to 19. One can see that the results are comparable to the simulation results shown in Figures 11 to 13; the fact that there are more valve actions in reality than in simulation can be explained by some leakages in the muscle and tubing and deviations due to parameter estimation errors. The desired pressure slopes are not as smooth as in simulation, this is mainly due to the noise on the input signal, most of it coming from the velocity part of the PID controller.

For more complex trajectories than a sine function the consumed energy will increase, because the imposed trajectory differs more from the natural movement of the pendulum, which is the sine function for small angles. The study of the sine wave is however of great importance for bipedal locomotion. The energy transfer mechanism used in walking is often referred to as an “inverted pendulum mechanism”. This observation has led to the development of the so called “passive walkers” where walking is a succession of swing motions. The attained 2.0 Hz is sufficient for walking bipeds, at higher speeds humans switch from walking to running and the preferred strategy to conserve energy becomes storing and releasing energy in elastic tissues. Because the movement of the center of mass during running is similar to a bouncing ball, running is often referred to as a “bouncing gait”. So during running of the biped another strategy has to be developed to reduce energy consumption than the one described in this paper. However, it is possible to increase the frequency above 2.0 Hz, which means the stiffness and thus also p˜ s has to increase.

5. Conclusion Actuators with adaptable compliance are gaining interest in the field of robotics. Little research, however, has been carried out on how to control the compliance. In this paper a strategy is proposed to control the compliance continuously in order to reduce valve switching and energy consumption by fitting the compliance of the actuator to the natural compliance of the desired trajectory. In this way the system dynamics are matched to the imposed trajectories. The effectiveness of the proposed control strategy has been shown through simulations and experimental results on a pendulum setup. The pendulum is actuated by an antagonistic pair of Pleated Pneumatic Artificial Muscles. By doing this both the torque and compliance are controllable. We believe the proposed


2.4

2.4

2.3

2.3 Pressure (bar)

Valve action


2.2 2.1 2 1.9


2.2 2.1 2 1.9

0

5

10 Time (s)

15

20

4.5

4.75

5 Time (s)

5.25

5.5

2.8

Pressure (bar)

Valve action

Fig. 17. Valve action and detail of the pressure changes for p˜ s = psoptimal (real).

2.7 2.6 2.5


2.8 2.7 2.6 2.5

0

5

10 Time (s)

15

20

4.5

4.75

5 Time (s)

5.25

5.5

2

2

1.9

1.9 Pressure (bar)

Valve action

Fig. 18. Valve action and detail of the pressure changes for p˜ s = 1.5psoptimal (real).

1.8 1.7 1.6 1.5 1.4

Real pressure Desired Pressure Valve action

1.8 1.7 1.6 1.5 1.4

0

5

10 Time (s)

15

20

4.5

4.75

5 Time (s)

Fig. 19. Valve action and detail of the pressure changes for p˜ s = 0.5psoptimal (real).


5.25

5.5

357

358


strategy can also be extended to other compliant actuators where both the torque and compliance are controllable. Future work will be the implementation of the proposed control architecture on a double pendulum and in a second stage on the biped Lucy.

Acknowledgments The author B. Vanderborght is a PhD student with a grant from the Fund for Scientific Research, Flanders (Belgium)(FWO). Lucy has been built with the financial support of the Research Council (OZR) of the Vrije Universiteit Brussel.

References Bicchi, A., and Tonietti, G. 2004. Fast and soft arm tactics: Dealing with the safety-performance trade-off in robot arms design and control. IEEE Robotics and Automation Magazine 11:22–33. Collins, S. H., Ruina, A., Tedrake, R., and Wisse, M. 2005. Efficient bipedal robots based on passive-dynamic walkers. Science 18:1082–1085. Daerden, F., and Lefeber, D. 2001. The concept and design of pleated pneumatic artificial muscles. International Journal of Fluid Power 2(3):41–50. Hirose, M., Haikawa, Y., Takenaka, T., and Hirai, K. 2001. Development of humanoid robot ASIMO. In Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems. Hurst, J. W., Chestnutt, J., and Rizzi, A. A. 2004. An actuator with physically variable stiffness for highly dynamic legged locomotion. In Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, USA, pp. 4662–4667. ISO (International Organization for Standardization). 1989. ISO 6538 Pneumatic fluid power—method of test— determination of flow rate characteristics of components using compressible fluids. Migliore, S. A., Brown, E. A., and DeWeerth, S. P. 2005. Biologically inspired joint stiffness control. In Proceedings of the 2005 IEEE ICRA International Conference on Robotics and Automation, Barcelona, Spain, April 2005, pp. 4519–4524. Pratt, G. A., and Williamson, M. M. 1995. Series elastic actuators. In Proceedings IEEE-IROS Conference, Pittsburg, USA, pp. 399–406.

Pratt, J. E., and Pratt, G. 1998. Exploiting natural dynamics in the control of a planar bipedal walking robot. In Proceedings of the 36th Annual Allerton Conference on Communication, Control, and Computing, Monticello, Illinois, pp. 739–748. Rogers, G., and Mayhew, Y. 1992. Engineering Thermodynamics, Work and Heat Transfer. New York, USA, John Wiley & Sons. Schulte, H. F. 1961. The characteristics of the McKibben artificial muscle. In The Application of External Power in Prosthetics and Orthotics. Publication 874, pp. 94–115, Lake Arrowhead: National Academy of Sciences–National Research Council. Slotine, J. J. E., and Li, W. 1991. Applied Nonlinear Control. Cambridge, MA, Prentice-Hall. Sulzer, J., Peshkin, M., and Patton, J. 2005. Marionet: An exotendon-driven, rotary series elastic actuator for exerting joint torque. In International Conference on Robotics for Rehabilitation (ICORR), Chicago, IL. Yamgushi, J., Nishino, D., and Takanishi, A. 1998. Realization of dynamic biped walking varying joint stiffness using antagonistic driven joints. In Proceedings of the IEEE International Conference on Robotics and Automatisation, Leuven, Belgium, pp. 2022–2029. Yokoi, K., et al. 2003. Humanoid robot’s applications in HRP. In Proceedings of the IEEE International Conference on Humanoid Robots, Karlsruhe, Germany. Van Ham, R., Verrelst, B., Daerden, F., Vanderborght, B., and Lefeber, D. 2005. Fast and accurate pressure control using on-of valves. International Journal of Fluid Power 6:53– 58. Vanderborght, B., Verrelst, B., Van Ham, R., and Lefeber, D. 2006. Controlling a bipedal walking robot actuated by pleated pneumatic artificial muscles. Robotica, in press. Verrelst, B., Van Ham, R., Vanderborght, B., Vermeulen, J., Lefeber, D., and Daerden, F. 2005. Exploiting adaptable passive behaviour to influence natural dynamics applied to legged robots. Robotica 23(2):149–158. Verrelst, B., Vermeulen, J., Vanderborght, B., Van Ham, R., Naudet, J., Lefeber, D., Daerden, F., and Van Damme, M. 2006. Motion generation and control for the pneumatic biped “lucy”. International Journal of Humanoid Robotics (IJHR), in press.


Exploiting Natural Dynamics to Reduce Energy Consumption by

Exploiting Natural Dynamics to Reduce Energy Consumption by

Suggest Documents

Exploiting Natural Dynamics to Reduce Energy Consumption by ...

Exploiting a Computation Reuse Cache to Reduce Energy in Network ...

How can Gorbyte Reduce Global Energy Consumption?

Exploiting V2G to optimize residential energy consumption with

Using consumer behavior data to reduce energy consumption

Using consumer behavior data to reduce energy consumption ... - arXiv

Ubi-PowerMeter: A Novel Paradigm to Reduce Energy Consumption

an approach to reduce energy consumption in cloud data centers

can organic farming help to reduce national energy consumption and ...

can organic farming help to reduce national energy consumption and ...

Exploiting Data Similarity to Reduce Memory Footprints

Adaptations by herbivores to natural enemies exploiting prey ...

Modifying Alcohol Consumption to Reduce Obesity ...

Exploiting Natural Resources - ANU

Exploiting Natural Dynamics in the Control of a 3D ... - CiteSeerX

Exploiting Natural Dynamics in the Control of a Planar ... - CiteSeerX

Exploiting Natural Resources - ANU

A Novel Approach for Reduce Energy Consumption in Mobile Cloud ...

Reduce Energy Consumption in Ad Hoc Network with Wireless Power ...

how can mobile applications reduce energy consumption? - QUT ePrints

Natural resource consumption caused by Finnish households

How to reduce your energy use by BOGE Compressors

How to reduce your energy use by BOGE Compressors

Natural Methods to Reduce Belly Fat