Are parallel manipulators more energy efficient

Are Parallel Manipulators More Energy Efficient? Yan Li and Gary M. Bone* Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada, L8S 4L7. *Correspondingauthor. E-mail [email protected] chines and optimal parameter selection, while Gregorio, Ahmadi and Buehler [Z]describe the design and control of an energy efficient monopod. To our knowledge no researchers have studied the influence of the mechanism type used for a manipulator on its energy efficiency. The objective of this paper is to compare the energy efficiency (m terms of the electrical energy usage) of a parallel manipulator to a serial manipulator with the same drive motors and a similar workspace. Tbe effects of end-effector position, velocity and acceleration, and static loading due to gravity, will be investigated.

Abstract The energy efficiency of a robotic manipulator is impoltant, particularly when that manipulator is used in conjunction with a mobile robot with limited battery life. In this paper the energy efficiency (io terms of the electrical energy usage) of a spatial three DOF parallel manipulator is compared to a serial manipulator with the same drive motors and a similar workspace. The effects of endeffector position, velocity and acceleration, and static loading due to gravity are examined. Over a range of conditions, the average energy usage of the parallel manipulator was determined to be 26% of the serial manipulator’s. This benefit is not due simply to the reduction in moving mass achieved by the parallel design since its moving mass is 70% of the serial manipulator’s. Static loading due to gravity was found to roughly double the power usage of both manipulators without significantly affecting their relative energy efficiency.

2. D.C. Motor Efficiency Before studying the manipulators’ power usage it is useful to examine the efficiency of the D.C. motors which are used with most robots. With a D.C.motor, the armahue current is proportional to the motor torque:

, =- r

(1) K. Where I‘ is the motor torque and K , is the torque coustant. The motor armature voltage is: dia V. = K,9. + L. + Rei. (2)

1. Introduction

e

As pointed out by Carlisle [I] at ICRA 2000, today’s industrial robots are far less efficient than human beings, and any improvements in manipulator energy efficiency would be both economically and environmentally beneficial. Also, when a robotic manipulator is used in conjunction with an untethered mobile robot (either to interact with the environment or to assist in locomotion) its energy efficiency is crucial to extending battery life. Previous work related to the energy efficiency of robots has focused on either path planning of conventional manipulators or gait planning for walking machines. Mayorga, Wong and Ma [SI describe an efficient local path generation approach for redundant or nonredundant manipulators which minimizes energy while avoiding both obstacles and singularities. Lee and Yamakawa [3] present a method for planning a minumum-energy collision free path which also maximizes the available dexterity. Marhefka and Onn [ 4 ] and Silva and Machado [ 7 ] analyze the problem of energy efficiency in walking ma-

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dt

Where

e, is the motor velocity, Ks is the back e m f con-

stant, L. is the armature inductance and R. is the armature resistance. The instantmeous motor power usage can be calculated using:

v,

Pd-, = x i, The mechanical power output is:

(3)

(4) prneCh =rxe, The efficiency of the motor can be defined as the ratio of the power output to the power input, i.e. Pme,,‘Pe,e,. Efficiency curves for a D.C. brushless motor are plotted in Figure I. Each cuwe is for a constant value of i. and r. The parameters for this particular motor are: Kb=7.5 Vikrpm, K,,,=0.06 N d A , L . 4 . 8 mH and R.=1.0 ohm.

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Clearly the efficiency is a function of both the motor velocity and torque. The efficiency is zero at zero rpm since electrical power is still consumed by the armature resistance. Manipulators with different kinematic structures require different joint velocity and torque profiles to produce the same end-effector motion and therefore will consume different amounts of electrical power if they use D.C. motors. The results presented in section I reflect this fact.

0.8,

End-effector

t

/ [ F i a = 1A ia= 2 A \ia

=3A

(r = 0.06 Nm)/

( r = 0.12Nm)

( r = 0.18 Nm) Figure 2. Kinematic structure of the University of Maryland parallel manipulator [9].

I

0

500

1000 1500 Velocity (rpm)

2000

Figure 1. Efficiency vs. velocity for a D.C. motor.

3. Manipulator Selection Typically, manipulator end-effector motions are primarily translational rather than rotational. Indeed, for many tasks three translational degrees-of-keedom (DOF) are sufficient. For these reasons we will select manipuY and Z) lators which provide three translational DOF only. To allow a fair comparison both will utilize only revolute drive motors. Based on the above criteria, the chosen parallel manipulator is the University of Maryland manipulator developed by Tsai and Stamper [9]. Its kinematic structure is shown in Figure 2. This manipulator is similar to the DELTA robot developed by Pierrot, Reynaud and Fournier [6] and to the "Flexpicker" recently introduced by ABB (81. The chosen serial manipulator is a standard three DOF articulated arm whose kinematic structure is shown in Figure 3.

e,

4L Yo

were assumed to be concentrated at their midpoints and the payload was modelled as a point mass located at the end-effector. The fust joint torque is given by:

4. Power Consumption for the Serial Manipulator Given the position, velocity and acceleration of the end-effector, the joint angles, velocities and accelerations may be obtained from the standard inverse kinematic equations. The manipulator dynamics were derived using the Lagrangian formulation. The masses of links 2 and 3

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Base Figure 3. Kinematic structure of the articulated serial manipulator.

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Where C,=cos(OJ, Cv=cos(8;+O$,Si=sin(OJ, Sij=sin(6j+6J, mi is the mass of the ith link and rn, is the mass of the payload. The second joint torque is: rl = [ ( ~ + m , + r n ~ ) u ~ + ( m , + 2 r nrn ~ ~ * o ~ ~ ~ + ( l 3 3

Where (py p,, pJ is the position of the end-effector (also termed the moving platform), cpi=cos(pJ,spi=sin(pJ and cO,=cos(OJ. Solving this set of equations yields the La+grange r n ~ multipliers ) a ~ ~ ~ I,,,I2 and ,I3.

rn rn rn +[(A+rnP)a,a,c,+ ( l + r n , ) a : @ ,+ [ ( L + r n ,

2

3

1 +rn,)a:c,s,+-(rn,

3

rn

+2rn,~,o,s(20,+e3)+(' 3

2

rn

+ rn,)a:c,,s,@: - (rn, + 2rn,)a2a,s,$B, - (2

(6)

2

rn +rn,)n,a,s,@ -'ga,C,

2

1 -rn,g(q~, + - a , ~ , , ) 2

- mng(a,C,+ o,Cd The third joint torque is: 7,

m,

rn

= [ ( ~ + r n , ) a , a , c+(-+rn,)a:16~ , 2 3

rn +(L+rn~a$~

3

1 +[-(m, + 2mn)o,a,~,s,,+ (2+ m p ) o Z , s , , @ :

2

+-(rn, 1

3 +2mP)a,a,s,8: - ( - +m, rnp)gn,~, 2

(7)

Figure 4. Definition of the joint angles [9].

r = J,&, + B,B, + r l R, (8) Where R, is the gear ratio ,J, = J, + J, is the sum of the

The driven joint torques are then obtained from: 7,

motor and gear inertia, and r is the joint torque. The motor annature current, voltage and instantaneous power usage are given by equations 1,2 and 3, respectively. The average absolute power usage for the manipulator is then: p.

3 I T "4 0

=ZFJIeklJdt

(9)

.. 1 + rn,a2)8,, +(-me + rn,)goce,,

2 - W [ @ , c + , + P A +h-r)se,, - P . c ~ , , I

3

1 7, = (-rnpz 3

(13)

.. + (-ma 1 + rn,a*)8,, + rn,)goc e,, 2

- 2 4 [ @ , c + , +P&

+h-W,,

-~,ce,,l

(I4)

.( +(-me 1 + rnba2p',, + rn,)gacO,, 2

- 2 4 [ ( +P,s+> ~ ~ +h-+@,, - P , C @ ~ , I (15) The average absolute power may then be calculated from the driven joint torques, velocities and accelerations using equations 1-3 and 8-9.

Manipulator The joint angles, velocities and accelerations are obtained from the end-effector position, velocity and acceleration using the inverse kinematics equations presented in [9]. The joint angles for each leg are defined as shown in Figure 4. The set of three constraint equations for the manipulator is:

6. Manipulator Design Parameters

2 ~ ? ( ~ , + h ~ , - r ~ - a ~ , c e , ~ ) = ( m , + r n ~ + 3(10) %)P, ,.I

-a+wo=(m, +mn +3%)Py

3

T~ = (-nip'

5. Power Consumption for the Parallel

/.I

1 = (-mea'

1

Where n is the motor number and T i s the consumption period. Note that if the energy consumed needs to be calculated it is simply the product of Paand T.

2x~b. +W-w?

(h) Side view

(a) Front view

2 For each joint, the motor torque is expressed hy the following equation:

(1 1)

2 ~ ~ , b ~ - a e , , ) = ( r n ~ + 3 ~ ) P ~ + ( r n , + q +(12) 3rn,)g

*

A3

To produce a fair comparison, the serial and parallel manipulators will be designed to have approximately the same sized workspace. The desired workspace is a sphere 1.0 m in diameter, centered at (0,0,0.75). As is typical in commercial serial robots we chose a p a 3 , so to meet the workspace requirement for the serial manipulator: a,=aJ=0.25 m and d,=0.5 m. The cantilever-like strncture of the serial arm requires bulky links to provide adequate stiffness, and the joint motors are typically contained within the links, so link masses of r n ~ m ~ =kg 5

were assumed. A substantial payload of m,=IO kg was also assumed. Two designs for the parallel manipulator were investigated. The first design has dimensions similar to those of the University of Maryland manipulator with a=kO.5 m, hz0.2 m and ~ 0 . m. 3 The second design has dimensions similar to the ABB Flexpicker robot, with a=0.35 m, 3 Both designs have a k 0 . 7 m, h=0.1 m and ~ 0 . m. workspace which is tent shaped and of approximately the same size as that of the serial manipulator. The motors for the parallel manipulator are attached to the fixed base. In additioq its truss-like structure provides stiffness even with slender links, so masses of m.=0.5 kg, mb=0.25 kg and mh=l kg were assumed. The same payload mass as the serial manipulator was employed. A standard brushless DC motor linked to a gearbox with a reduction Rg=200was selected for all driven joints. The motor parameters are: K ~ 7 . 5Vikrpm, K,,,=0.06 NdA,L.=0.8 mH, R,=I.O ohmand Jm=5.2e-5kgm2.

will be calculated to determine the significance of gravity on the energy efficiency of each manipulator. 1.2 Results and Discussion Power ratio results for the first parallel manipulator design are summarized in Table 1. The parallel manipulator is on average more energy efficient for the X and Y direction moves (since their mean PR values are greater than one), however it is less energy efficient for the Z moves. There are also several minimum PRvalues which are much less than one, particularly for cases 3-4 and 7-12 which means the parallel manipulator is very inefficient in certain positions. The ‘% of P p l ” metric measures how often the parallel manipulator is more energy efficient for the given power ratio surface. For cases 5 , 7 and 11, the values are less than 50% so the serial manipulator is more energy efficient at a greater frequency than the parallel manipulator. The second parallel manipulator design proved to he superior to the first. All of the remaining results in this section are for the second design. Its power ratio results are summarized in Table 2. In all but cases 4, 11 and 12 its power ratio was always greater than one. The minimum PRvalues are also close to one in all cases so the parallel manipulator never used significantly more energy than the serial manipulator. As with the fust design, the PRvalues are smaller for moves in the Z direction, particularly when gravity is in effect (cases 5 and 1 I). The average of the mean PR values kom Table 2 is 3.8. In other words, the parallel manipulator consumed on average only 26% of the energy of the serial manipulator.

7. Power Comparison 7.1 Procedure The methods outlined in Sections 4 and 5 are used to calculate the Pavalues for the manipulators. To provide a means of comparison we define the power ratio as: Serial Pa (16) PR = Parallel Pm Whenever P p l the parallel manipulator is more energy efficient than the serial one and vice-versa when P R 4 . The power consumed by each manipulator will vary as a function of its position, velocity and acceleration. To limit the scope, we assume that the manipulator endeffector is to follow a linear trajectory in the direction of one of the coordinate axes (X, Y or Z), starting at rest, with a period of constant acceleration followed by an equal period of constant deceleration. Pa and PR values will be calculated for a range of accelerations and average velocities. To study the effect of position, the power ratio was calculated for small motions taking place at equally spaced points on a plane through the common workspace of the manipulators to produce a ‘power ratio surface”. More specifically, for the planes Z=0.75 m and X=O which slice through the middle of the workspace P, values were calculated at 0.02 m intervals for 0.04 m moves with acceleration=deceleration=4.5 d s ’ . A fwther issue is the effect of static loading due to gravity. For space applications or with static balancing (including the payload) the gravitational constant is effectively zero. Pnand PR values with g=O and ~ 9 . d8s ’

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1.4e-3 12 Y 0 1.92 84.8 11 Z 9.8 5.6e-5 28 0.99 29.4 12 I z I 0 3.4e-4 I 131 I 2.07 I 82.5 I I Table 1. Power ratio results for fust parallel manipulator design. ‘Cases 1-6 are for the Z=0.75 m plane and cases 1-12 are for the X=O plane.

,

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PR results, it was determined that the power ratio was purely a function of the motion path and not a funotion of either the velocity or the acceleration.

...: . . '. ....

nipulator design. *Cases 1-6 are for the 2=O.75 m plane and cases 7-12 are for the X=O plane. Figure 5. Power ratio surface for X direction movement in the plane Z=0.75 m with gravity. (Note: To improve image contrast PRvalues greater than 5 are not shown.)

Figures 5 and 6 give a more detailed view of the variation of the power ratio over the 2=0.75 m slice through the workspace for cases 1 and 5, respectively. Figure 7 shows the power ratio surface for the case 10, X=O slice. In general the serial manipulator consumed more energy at the center of the workspace and near its edges. The parallel manipulator had a relatively constant energy usage which only increased when it approached the edge of the workspace. As the result, the power ratio surfaces displayed larger values near the workspace center and edges. These appear as lighter bands in the figures. A similar pattern was observed with the X=O slices through the workspace. In Figure 5, smaller PR values can be observed near X=O, Y=+0.3 m. In these locations the first joint is not needed for the serial robot to provide movement in X so its total power usage is reduced. Moves of 0.3 m in X, Y and 2 with accelerations fiom 0.6 to 4.8 m/s2 were used to examine the effect ofvelocity and acceleration on power consumption and the power ratio. Given that mechanical power equals the product of force and velocitv. and that force equals mass times acceleration, a reasonable hypothesis is that the average power will increase linearly with the product of acceleration and average velocity. This hypothesis held hue for both types of manipulator when the gravity effect was zero, as shown in Figure 8. However, with gravity in effect a different relationship was observed. The gravity force was predominant, so the acceleration had little effect, and the power consumption varied almost linearly with the average velocity, as shown in Figure 9. Comparing the respective Pa values from the two figures it can be observed that the presence of gravity (or the absence of static balancing) caused the power usage to roughly double. From the corresponding

..

.

.

.

n

c

Figure 6. Power ratio surface for 2 direction movement near the plane z=o.75 with gravity. (Note: T~ improve image contrast PRvalues greater than 1.5 are not shown.)

...:'...'....... . ... .. ..

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Figure 7. Power ratio surface for Y direction movement near the plane X=O without gravity. (Note: To improve image contrast PRvalues greater than 5 are not shown.)

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. . . -1..

........

......... .... ’ . . .

0.2

1.2

....

2.2

3.2

4.2

5.2

Rcduct of Aweletam and Averags V e a i (+UsAS)

A F a d e l Z m e . x Wi. Z m e . D Famld X m e .

%MI, X m

e.

Figure 8. Average absolute power consumption (Pa) versus average velocity without gravity vs. product of end-effector accelerationand average velocity.

parallel manipulator design. It is also space inefficient in that it must be twice as large as the serial manipulator (i.e. a+b=2.(aa+a,) ) to produce the same workspace. The results also demonstrated that the presence of gravity (or the absence of static balancing) caused the energy consumption to roughly double. With gravity in effect, the power usage increased almost linearly with the average velocity, while without gravity it increased linearly with the product of acceleration and average velocity. There are some limitations of our analysis. Losses such as joint fiction, amplifier and gearbox inefficiency were not included. These would cause the power usage of both manipulators to increase, so our P. values underestimate the real situation. However, since the same motors were used with both robots, their usage increase should be similar and the relative energy efficiency should not change significantly.

N’m 5

150

5 1 120 40

. HE 1w 80 3

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References

................... . ~. . ./ .: .:.:.:. . . . . . . . . . . . . . . .

1. B. Carlisle. Robot Mechanisms. IEEE Int. Conz on Rob. &Auto., 701-708,2000.

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*

.........

40

. . .

......

20

......

...............

2. P. Gregorio, M. Ahmadi and M. Buehler. Design, Control and Energetics of an Electrically Actuated Legged Robot. IEEE Trans. on Syst., Man. and Cyber., 27, 626-634. 1997.

~~~~

3. S.-J. Lee and H.:Yamakawa. Study on MinimumEnergy Collision-Free Trajectory Planning for Rigid Manipulators. Trans. of the JSME, Part C.. 63, 20302036,1997.

4. D.W. Marhe&a and D.E. Orin. Gait Planning for Energy Efficiency in Walking Machines. IEEE Int. Conf on Rob. &Auto., 474-480, 1997.

Figure 9. Average absolute power consumption (P.) with gravity in effect vs. average end-effector velocity.

5 . R.V. Mayorga, A.K.C. Wong and K.S. Ma. Efficient

8. Conclusions

Local Approach for the Path Generation of Robot Manipulators. J. o f l o b . Syst., 7,23-55, 1990.

For the three DOF manipulators compared in this paper the answer to the question posed by the paper’s title is a definitive “Yes!” Over a range of end-effectors positions the average energy usage of the parallel manipulator was only 26% of the serial manipulator’s. This higher efficiency was not significantly affected by end-effector velocity, acceleration or static loading due to gravity. This benefit is not simply due to the reduction in moving mass achieved by the parallel manipulator since the ratio of moving masses @aralleVserial) is 0.7. However the parallel manipulator is not without drawbacks. It must be properly designed to achieve the improved efficiency as evidenced by the relatively poor performance of the first

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6. F. Pierrot ;C. Reynaud and A. Foumier. DELTA: A Simple and Efficient Parallel Robot. Robotica, 8, 105109,1990. 7. F. Silva and J.A.T. Machado. Towards Efficient Biped Robots. IEEE Int. con^ on Intel. Rob. & Syst., 394-399, 1998.

8. R. Spencer. Robots Help Packaging lndushy Keep Up with the Times. Robotics World, 26-29, Jan/Feb 2000. 9. L.-W. Tsai. Robot Analysis: The Mechanics of Serial and Parallel Manipulators. Wiley-Interscience, 1999.

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