computed-torque control of a four-degree-of-freedom ... control of a four-degree-of-freedom...

Computed-torque control of afour-degree-of-freedom admittancecontrolled intelligent assist device

Alexandre Lecours and Clement GosselinDepartement de genie mecanique, Universite Laval, Quebec, Qc, Canada

Abstract Robots are used in different applications to enhance human per-formance and in the future, these interactions will become more frequent. Inorder to achieve this human augmentation, the cooperation must be very intu-itive to the human operator. This paper proposes a computed-torque controlscheme for pHRI using admittance control. The admittance model is firstintroduced. Then, the robot identification, the computed-torque approachand the saturation considerations are addressed. The intelligent assist deviceused for the experiments is then presented. Finally, experimental results thatdemonstrate the performance of the algorithm are provided.

1 Introduction

The main challenge for human augmentation systems is to perceive theirenvironment and the human intentions and to respond to them adequately,intuitively and safely [7]. To this end, it is desired to enhance the controlperformances of such systems. On the other hand, computed-torque schemesare widely used in robotics for reference trajectory following since they de-couple and linearize highly non-linear dynamical systems, thereby leading tobetter performances than simple PID control [4, 12, 13, 21, 23, 24, 6]. Suchtechniques are also used in haptics to improve impedance control performanceand virtual environment rendering [9, 17, 18]. Computed-torque schemes werealso very briefly introduced for admittance control in [8]. Admittance controlis typically used for applications involving physical human-robot interactions(pHRI) with large payloads. When admittance control is used, a handle or aforce/torque sensor is normally used to detect human intentions [16, 5].

With reference trajectory following, it is possible to design a smooth ref-erence trajectory in order to avoid discontinuous acceleration or jerk profiles.However, obtaining smooth reference trajectories is less straightforward in

1

Proceeding of the 13th International Symposium on Experimental Robotics,Quebec, QC, Canada, June 17–21, 2012.

2 A. Lecours and C. Gosselin

Fig. 1 Four-dof intelligent assist device prototype used in the experiments.

human-robot collaboration since the trajectory is directed by the human op-erator. This paper presents a methodology to overcome these issues, alongwith experimental examples and results. Although the robot used in the ex-periments is decoupled and linear, it is shown that the computed-torque con-trol leads to better performance. It is then possible to implement the controlalgorithms on a nonlinear robot where the performance improvement overPID control should be even greater.

This paper proposes a computed-torque control scheme for pHRI. The pa-per is structured as follows. The admittance model is first introduced. Then,the robot identification, the computed-torque approach and the saturationconsiderations are addressed. The intelligent assist device used for the exper-iments is then presented. Finally, experimental results that demonstrate theperformance of the algorithm are provided.

2 Admittance model

Two main classes of control schemes are used in haptic applications and pHRI,namely, impedance and admittance control. Because of the large inertia andsignificant friction of the intelligent assist device (IAD) used in this work, itwould obviously be too hard for a human operator to impart a movementto the IAD, which makes impedance controllers not well adapted for thesituation, even if a force sensor is used. An admittance controller is thus usedas detailed in [19].

The one-dimensional admittance equation is written as:

fH = mx+ cx. (1)

Computed-torque control of a 4-dof admittance controlled IAD 3

where fH is the interaction force, i.e., the force applied by the human oper-ator, m the virtual mass, c the virtual damping and x, x, x are respectivelythe position, velocity and acceleration.

The trajectory to be followed by the robot can be prescribed as a positionxd or as a desired velocity xd. For velocity control, the desired velocity canbe written, in the Laplace domain, as:

Xd(s) =FH(s)ms+ c

=FH(s)/cmc s+ 1

= FH(s)H(s). (2)

where Xd(s) is the Laplace transform of xd, FH(s) is the Laplace transformof fh and s is the Laplace variable. Velocity control is used here, similarly towhat was done in [19, 11, 25].

3 Model identification

In order to use computed-torque control, a dynamic model of the robot isneeded and is obtained using model identification techniques. In an identi-fication scheme, a given variable is used as the dependent variable and theothers as independent. The coefficients applied to the latter are the parame-ters to be identified.

3.1 Model

The dynamic model of the robot is:

τ = M(q)q + C(q, q)q + g(q) + τf (3)

where q is the vector of joint displacements, M(q) is the generalized inertiamatrix, C(q, q)q is the vector or centripetal and Coriolis effects, g(q) is thevector of gravitational effects and τf is the vector of joint friction.

Because the system used in the experiments is decoupled, each dof is iden-tified separately for simplicity and only the X and Y axes are used in theexperiments. The discrete-time equations are:

u(k) = mI x(k) + τvI(k) + τcI(k) (4)

where u is the command, expressed as a current with units of A, at time stepk, mI is the inertial term expressed at the motor in As2/m, τvI and τcI arerespectively the viscous and Coulomb friction and are simply modeled hereas:


τcI = fcIsign(x)τvI = fvI x (5)

where fcI is the Coulomb friction coefficient, expressed in A, and fvI is theviscous friction coefficient, in As/m. Although a more complex friction modelcould be used such as in [26, 3, 1, 14, 15], the chosen model is sufficient hereas it will be shown in Section 4.3.

Defining:

y(k) = u(k)φt = [x(k), x(k),−sign(x)]T

θt = [mI , fvI , fcI ]T (6)

the model is written as:

yt(k) = φtTθt + ey(k) (7)

where ey stands for the error. Friction parameters could also be identifiedseparately, as in [14, 15].

To solve this problem, simple least squares [14, 13, 20], recursive leastsquares [20] and recursive least squares with approximate maximum-likelihood[20] can be used. The latter technique was used in the experiments and sincethe parameters are identified off-line, the data is filtered with a non-causalFIR filter to obtain better convergence properties without creating phaseshift. The velocity and acceleration are obtained from the position with atwo-point Lagrangian derivative centred on the current point. The data isalso normalized to obtain better numerical stability.

3.2 Results

The results obtained from the identification are first compared with the mea-sured values and presented in Tab. 1. The mass was measured by summingall the component masses and by adding the motor inertia, transferred at theend effector. The Coulomb friction was measured by manually pushing thedevice with a dynamometer at a very low velocity. It should be noted thatthe comparison is an approximation aiming at providing ballpark figures.

Fig. 2 presents the comparison between the command signal used for theidentification and the reconstructed command obtained with the identifiedparameters. The velocity and acceleration are filtered with a non-causal FIRfilter to remove high-frequency noise in order for the comparison to be possi-ble. The estimated torques do not match exactly the applied torques since themodel is a simplification of reality but they are nevertheless fairly accurate.


Table 1 Identification results.

Mx τvx τcx My τvy τcy

(kg) (Ns/m) (N) (kg) (Ns/m) (N)

Measured 507 N/A 88 328 N/A 51

Identification 507 369 92 356 32 71

Figure 3 shows an open-loop response to a human input, obtained bysimply feeding the inverse dynamics to the four-dof assist device as:

τ = M(q)qd + C(q, q)q + g(q) + τf (8)

or in this simple case:

u(k) = mI xd + τvI xd + τcI . (9)

More details on the implementation of this control are given in Section 4(Subsections 4.3 and 4.4) for the friction compensation part. Even in anopen-loop mode, it is easy for the operator to cooperate with the robot. Thecondition number (ratio between the maximal and minimal singular values)of the regressor matrix is 4.2 for the X axis and 4.1 for the Y axis. It is desiredfor this number to be close to 1 (and it should be below 100 [10]), for theregressor matrix to be well-conditioned and the estimation of the parametersto be reliable [14, 10].

X axis

Y axis

Com

mand

(A)

Com

mand

(A)

Measured value

Estimated value

Time (s)

0

0

0

0

5

5

5

5

10

10

10

10

15

15

15

20

20

25

25

30

30

35

35

40

40

-5

-5

-10

-10

-15

Fig. 2 Offline measured and estimated

torque comparison.

X axis

Y axis

0

0

0

0

5

5

10

10

15

15

20

20

25

25 30

-0.6

-0.6

-0.4

-0.4

-0.2

-0.2

0.6

0.6

0.4

0.4

0.2

0.2

Vel

oci

ty(m

/s)

Vel

oci

ty(m

/s)

Time (s)

Measured velocity

Desired velocity

Fig. 3 Open-loop reference velocity

tracking in human-robot collaboration

mode.


4 Velocity controller

In this section, the PID and computed-torque control used for the low-levelvelocity controller are explained. Even if the dynamics of the robot usedfor the experiments are linear, it will be shown that the computed-torquecontrol leads to better results than the PID control. Additionally, it is possibleto implement the control algorithms with a nonlinear device, in which casethe performance improvements over PID control should be greater. Frictioncompensation is also discussed.

4.1 PID control

First, PID control is considered for the velocity controller. The output com-mand is:

τ = KP e +KDe +KI

∫e (10)

where e = qd−q, KP , KD and KI are respectively the proportional, deriva-tive and integral gain matrices, qd is the desired joint velocity vector and qis the measured joint velocity vector. This controller is applied to all jointsindependently. As pointed out in [19], it is not recommanded to use a deriva-tive gain since the signal is noisy [2] (acceleration signal) and no integral gainis used since the behaviour to an operator input would then depend on theerror history. This introduces additional limitations and leads to a lack offlexibility of the PID controller.

4.2 Computed-torque control

Computed-torque control is widely used in robotics. Its main advantage isto transform a complex nonlinear multi-input multi-output (MIMO) systeminto a very simple decoupled MIMO linear system [23, 4, 12, 13, 21, 24, 6].

Numerous different approaches exist for the implementation of computed-torque control. A very popular approach [4, 12, 21] is written as

τ = M(q)[qd +KP q +KV˙q] + C(q, q)q + g(q) (11)

where q = qd − q.The PD+ approach [12, 21] is written as

τ = M(q)[qd] + C(q, q)qd + g(q) +KP q +KV˙q (12)


while the non-adaptive version of the Slotine and Li controller [24, 23] iswritten as

τ = M(q)[qd +Λ ˙q] + C(q, q)[q +Λq] + g(q) (13)

+KP q +KV˙q

where Λ = KV−1KP .

These approaches were widely applied for trajectory tracking. However,velocity control is considered here. Eqn. (11) is then modified accordingly,similarly to what was done in [22]. This leads to

τ = M(q)[qd +KP˙q] + C(q, q)qd + g(q) (14)

where there is no derivative nor integral gains for the reasons explained inthe description of the PID controller.

The dynamics can also be only partially compensated for. The advantageof a partial compensation is to benefit from the advantages of the computed-torque technique while not relying too much on a model. It is also pointedout that the model part of the control is low-pass filtered in order to avoidhigh frequency command input.

4.3 Friction compensation

Friction compensation can be included in the controller as follows:

τ = M(q)[qd +KP˙q] + C(q, q)qd + g(q) + τc + τv (15)

where τc and τv are respectively the Coulomb and viscous friction vectorswhose their ith components are written as:

τ ic = f ic · sign(xid) · (1− e−αi|xi

d|) (16)τ iv = f ivx

id.

The exponential term along with the α parameter is used to reduce thechattering induced by friction compensation when the velocity is near zero.The desired velocity is used for viscous friction compensation in order toreduce the command noise and contribute to the command based on thehuman intention. A more complex model could have been used [26, 3, 1],including stiction for example, but as it will be shown, the simple frictioncompensation from eqn. (16) is sufficient in practice for the device used here.


4.4 Desired velocity and acceleration

Computed-torque is well adapted to classical position control trajectory fol-lowing since the desired trajectory is known and can be designed to besmooth.

With admittance control, the trajectory is known (see eqn. (2)) andcomputed-torque control can then be used by inputting the desired velocityand acceleration in eqn. (15). However, the trajectory relies on the opera-tor’s intentions, which may not lead to smooth signals. The smoothness canhowever be controlled by a proper choice of admittance parameters (virtualdamping and mass) and from force input signal pre-processing. Saturationmust also be considered, as detailed in the next section.

With impedance control, desired velocity and acceleration cannot be usedfor computed-torque since they are unknown. Measured signals can be usedbut this requires very good position, velocity or acceleration sensors andsignal processing.

5 Saturation consideration

For safety reasons, the desired velocity and acceleration should be limited.However, such limitations can lead to abrupt variations in the accelerationor jerk profiles, which is undesirable since such variations result in abruptvariations in the command. The problem is amplified with computed-torquecontrol since the command is directly related to the desired acceleration andvelocity. Applying such commands to the robot can excite unmodelled dy-namics and lead to vibrations.

5.1 Velocity limits

If a simple velocity saturation is used, the acceleration at a saturation pointwill go from a given value to zero within one time step, leading to an abruptacceleration profile. If the desired velocity is filtered, the desired accelerationwill be smoother but at the expense of time delays. In order to alleviatethis problem, the desired velocity is gradually limited as it approaches thesaturation limit, using a third or fifth order polynomial as shown in Fig. 4.The third order polynomial is obtained by setting the output velocity equalto the input velocity for input velocities of (vsat−δ/

√2) and (vsat+δ) (where

δ is a tuning parameter) and by setting the first derivative for these sameinput velocities to one and zero respectively. For the fifth order polynomial,the second derivative is also set to zero for the same input velocities.


Fig. 5 presents the desired velocity, acceleration and jerk response to ahuman input sine force. Only the results obtained with the third order poly-nomial are shown for simplicity. It is shown that the desired acceleration issmoother with the third order polynomial than with basic saturation. Thisis verified with the desired jerk which is approximately 270 m/s3 for basicsaturation and approximately 40 m/s3 with the third order polynomial. Thesaturation transition smoothness can be varied by changing the value of pa-rameter δ. In this example, the velocity limit is 0.7 m/s and δ is in the orderof 0.23.

3rd order polynomial

Basic saturation

5th order polynomial

Velocity in (m/s)

Vel

oci

tyout

(m/s)

00

0.5

0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.7

0.6

Fig. 4 Theorical velocity saturation.

No saturation

Basic saturation


Time (s)

Jerk

(m/s3

)A

ccele

rati

on

(m/s2

)Velo

city

(m/s)

0

0

0

0

0

0

0.5

0.5

0.5

0.5-0.5

1

1

1

1

1

-1

1.5

1.5

1.5

2

2

2

2

-2

2.5

2.5

2.5

100

-100

-200

-300

Fig. 5 Desired velocity, acceleration and

jerk with velocity saturation for a human

input sine force.

No saturation

Basic saturation


Time (s)

Jerk

(m/s3

)A

ccele

rati

on

(m/s2

)Velo

city

(m/s)

0

0

0

0

0

0

0.5

0.5

0.5

1

1

1

1

-1

1.5

1.5

1.5

2

2

2

2

2

-2

-2

3

3

3

4

-4

2.5

2.5

2.5

200

-200

400

600

Fig. 6 Desired velocity, acceleration and

jerk with acceleration saturation for a

human input sine force.


5.2 Acceleration limits

A common situation in which the acceleration can vary rapidly (other thana highly dynamic human force input) arises when the maximum allowed ac-celeration is not the same as the maximum allowed deceleration. However,this situation is desirable for safety considerations: it must not be allowedto increase speed very rapidly but one should be able to stop rapidly. Then,when the device goes from a deceleration phase to an acceleration phase,the desired acceleration changes abruptly from the maximum allowed decel-eration to the maximum allowed acceleration. Filtering could help but atthe expense of time delays, which is not desirable. Similarly to what is donefor velocity saturation, the desired acceleration is gradually limited using anexponential function with the desired velocity as a parameter. Indeed, thetransition occurs at a desired velocity of zero and the desired accelerationis then gradually varied as a function of the desired velocity. The desiredacceleration transition is then represented by

am =x+m + x−m

2sign(xd)− −x

+m + x−m

2sign(xd)

(1− e−|γxd|

)(17)

where γ is a smoothness parameter, am is the current maximum allowedacceleration/deceleration, x+

m is the maximum allowed acceleration and x−mis the maximum allowed deceleration. Parameter γ should be high enough toobtain smoothness but not too high since it affects the maximal accelerationlimit when the velocity is near zero.

Fig. 6 presents the results with a human input sine force. It is shown thatwith the gradually changing acceleration saturation, the transition is muchsmoother. This is verified with the desired jerk which is approximately equalto 500 m/s3for basic saturation while it is reduced to 30 m/s3 with theproposed transition from eqn. (17). A jerk limiter could also be implementedif the force signal is not too noisy.

5.3 Virtual limits

A very intuitive means of implementing virtual limits is to make the desiredvelocity zero if the position is greater than a given limit and the velocityis directed toward the limit. To avoid very large required acceleration, anacceleration limiter can be implemented although the acceleration and jerkprofile would remain abrupt. The proposed solution is to set the force to zero,with a rate limiter and with high and well chosen admittance parameters. Theadmittance parameters can also be increased according to the position and astiffness term can also be added.


6 Prototype of a 4-DOF intelligent assist device

The robot used for the experiments reported in this paper is a prototype ofa 4-dof intelligent assist device (IAD), shown in Fig. 1, allowing translationsin all directions (XY Z) and a rotation (θ) about the vertical axis. In thisprototype, the total moving mass is approximately 500kg in the directionof the X axis and 325kg along the Y axis. Additionally, the payload mayvary between 0 and 113kg. The horizontal workspace is 3.3m× 2.15m whilethe vertical range of motion is 0.52m. The range of rotation about the ver-tical axis is 120◦. Three different control modes are possible: autonomousmotion, unpowered manual motion and interactive motion (cooperation). Inthis paper, only the latter is addressed. The controller is implemented on areal-time QNX computer with a sampling period of 2ms. The algorithms areprogrammed using Simulink/RT-LAB software.

7 Experimentation

In order to demonstrate the effectiveness of the proposed control algorithms,three experiments were performed. The first one consisted in simply movingthe assist device to compare the PID and computed-torque velocity con-trollers. The second experiment is a drawing task and the third one consistedin asking the operator to trace imaginary circles in mid air.

7.1 Error and noise reduction

This experiment consisted in moving the intelligent assist device (see Fig. 1)back and forth to compare the magnitude of the error and the command noisebetween the PID and computed-torque velocity controllers. The parametersused in the PID controller are KPx = 0.05, KPy = 0.06, and KIx, KIy,KDx and KDy are zero as previously explained. The closed-loop gains usedin the computed-torque controller are mIxKPx = 0.04 and mIyKPy = 0.04while the inertial and friction terms were taken to be 90% of the identifiedparameters (Tab. 1). With the PID control, it is not possible to increase thegains significantly since vibrations or instability occur. With the computed-torque control, it is possible to adjust the inertial and friction compensationwhile it is also possible to adjust the closed-loop gains. By increasing thesegains, the error can be reduced, at the expense of command noise.

With the selected parameters, the error reduction from the PID controllerto the computed-torque controller is about 50% for the X axis and 32% forthe Y axis. The noise amplitude was approximately reduced by 20% for theX axis and by 33% for the Y axis. These results are shown in Figs. 7 and 8.


Fig. 9 compares both control laws for low velocity reference. It is clearthat the results are much better with the computed-torque controller andthis is clearly apparent to the human user as described in the subsequentexperiments.

Time (s)Time (s)

Command (A)

Velocity error (m/s)

Computed-torque PID

Velocity (m/s)

0

0

0

0

0

0

0

0

0

0

0

0

0.20.2

-0.2-0.2

0.50.5

-0.5-0.5

5

5

5

5

5

5

10

10

10

10

10

10

2020

-20-20

Fig. 7 X axis velocity error andcommand noise comparison between

computed-torque and PID control.

Time (s)Time (s)

Command (A)

Velocity error (m/s)

Computed-torque PID

Velocity (m/s)

0

0

0

0

0

0

0

0

0

0

0

0

0.20.2

-0.2-0.2

0.50.5

-0.5-0.5

5

5

5

5

5

5

10

10

10

10

10

10

10

10

-10-10

Fig. 8 Y axis velocity error and

command noise comparison betweencomputed-torque and PID control.

7.2 Drawing task

The drawing task consisted in asking the operator to trace a simple maze,fixed to the ground, (shown in Fig. 11) with a pen mounted on the IAD at 1.4metre from the operator. The instructions were to minimize the completiontime and the overshoots in the maze. Experiments were performed with thePID and computed-torque controllers. The admittance parameters were fixed(c = 60Ns/m and m = 36kg).

The experiment was performed by 6 subjects whose age ranged between 25and 41. Task completion time, maze overshoots and subject comments wererecorded. The subjects were allowed some practice before performing the task.Subjects were not told which control was set and the order was varied betweensubjects. Fig. 10 shows the task completion time along with the distance ofovershoots (the total length of the curve outside the maze). The completion


Vel

oci

ty(m

/s)

Vel

oci

ty(m

/s)

Measured velocity

Desired velocityComputed-torque

PID

Time(s)

0

0

0

0

0.05

0.05

-0.05

-0.05

0.1

0.1

-0.1

-0.1

20

20

40

40

60

60

80

80

Fig. 9 Y axis velocity tracking compar-

ison at low velocity.

Time (s)

Over

shoots

(cm

)

Computed-torque

PID

0

2

4

6

8

12

10

20 25 30 35 40 45

Fig. 10 Distance of overshoots along

with the time needed to complete the

drawing task. The larger markers repre-sents the group average.

time is similar while the overshoot distance is about 32% lower with thecomputed-torque control. Subjects reported it was easier to perform highacceleration or deceleration, that it was easier to change direction and thatthe feeling was better at low velocities with the computed-torque control. Thedifference should be even more noticeable when the algorithms developed inthis paper are applied to a more typical nonlinear robotic system for which theperformance obtained with the computed-torque technique is usually muchbetter than with PID control.

7 cm

0.9 cm

Fig. 11 Maze trajectory with an exam-

ple drawing.

Subject1

Subject2

Subject3

Subject4

Subject5

Subject6

Average

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Excentr

icity

PID

Computed-torque

Fig. 12 Comparison of the circle excen-

tricity.


7.3 Circles

The last task consisted in asking the operator to trace imaginary circles inmid air. The average eccentricity (ratio of the short axis over the long axis)of the circles was measured from the position data. The results are shownin Fig. 12. In average, the eccentricity was 0.78 with PID control and 0.89with computed-torque control, a 14% improvement. The subjects generallyreported that it was easier to perform great circles with computed-torquecontrol. Indeed, it is easier to change direction, especially because of frictioncompensation.

Conclusion

This paper presents computed-torque control adapted to admittance con-trol. The admittance model, the robot identification, the computed-torqueapproach and the saturation considerations are presented. Finally, experi-mental results demonstrate the performance of the algorithm on a full-scaleintelligent assist device prototype.

Acknowledgment

This work was supported by The Natural Sciences and Engineering ResearchCouncil of Canada (NSERC) as well as by the Canada Research Chair Pro-gram and General Motors (GM) of Canada.

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