The Application of Linear Conditioning to Maximize ...

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THE APPLICATION OF LINEAR CONDITIONING TO MAXIMIZE ACTUATOR AND HARMONIC DRIVE UTILIZATION ON A SINGLE-JOINT INDIRECT-DRIVE UNIT Erwin Satrya Budiman' Masayoshi Tomizuka"

• Berkeley Applied Science fj Engineering, Inc., USA 1431 Oakland Blvd. Suite 215, Walnut Creek, CA 94596 erwin. [email protected] •• University of California at Berkeley, USA

Abstract: This paper considers maximizing the utilization of actuator and Harmonic Drive (HD) of a single-joint indirect-drive unit. To achieve this, a Linear Conditioning (LC) structure is added on an existing controller. LC, which considers the control of linear systems with bounded input, is used here to address both actuator saturation and maximum allowable HD torque limits. For poorly damped systems, vibration damping is added to reduce the constraint caused by the maximum allowable HD torque. In this case, additional adjustment to the saturation limit of the LC is made. Copyright © 2002 IFAC Keywords: Input saturation, torque limitation, Harmonic Drive, Linear Conditioning

because a given actuator input may result in joint torque levels that exceed the allowable HD torque. There are two extreme cases. The first is when the actual limit is much smaller than the virtual limit. From a physical design point of view, the first case implies that either the HD is stronger than necessary, or the actuator is not powerful enough. From a control design point of view, the actuator limit is the constraint to consider. The second case is the opposite. From a physical design point of view, the second case means that either stronger HD is required, or a less powerful actuator may suffice. From a control design point of view, joint damping can increase the virtual limit by reducing joint resonance. In addition, the virtual limit must be considered as the constraint of the actuator. In both cases, saturation remains an issue.

1. INTRODUCTION

Many control laws assume that the actuator has no amplitude constraint. To satisfy this assumption, actuators may be specified to be more powerful than necessary. An overly powerful actuator may not be optimal from the physical design point of view. In addition, increasing actuator power usually increases its size and inertia. The larger inertia may require other mechanical parts to be more massive. From these viewpoints, minimizing actuator size for a given payload is beneficial. However, a smaller actuator usually has a lower actuator saturation limit. This can be partially alleviated by the use of Harmonic Drive (HD) gearing. A large gear ratio allows the use of high-speed, low-torque actuators, which are typically significantly smaller than low-speed, high-torque actuaton;. Nevertheless, the resulting Indirect-Drive unit may still have actuator saturation issues. In addition to this actual limit, a virtual limit may be imposed by the HD on the actuator. This is

Many saturation compensation approaches for linear systems with input saturation can be categorized into Linear Conditioning (LC). The term linear in LC implies that other than the nonlin-

427

ear saturation model, the rest of the LC structure is linear. Likewise, except for the input saturation, the plant is assumed to be linear. Practically, LC involves the addition of linear transfer function(s) that utilize a combination of the demanded actuation and the achieved actuation. Relevant works on LC can be found in (Campo & Morari, 1990), (Hanus, Kinneart, & Henrotte, 1987), (Hippe & Wurmthaler, 1999), (Weston & Postlethwaite, 2000), and (Kothare, Campo, Morari, & Nett, 1994), among others.

l I

" z=\.'

Fig. 1. System with LC

Many existing LC schemes can be interpreted in terms of Kothare's unified conditioning (Kothare, Campo, Morari, & Nett, 1994). Weston and Postlethwaite demonstrated the equivalence of their approach to that of Kothare's unified conditioning. Additionally, Weston and Postlethwaite presented a particular LC structure which allows saturation to be interpreted as a nonlinear disturbance to the output of the closed-loop (nominal) linear plant.

Fig. 2. Conceptual diagram for Le Assuming that Cmode.l is identical to C, and that M is nonzero, Fig.1 can be shown to be conceptually equivalent to Fig.2. In Fig.2, the system is separated into 3 parts: an intended linear system, a nonlinear loop, and a disturbance filter. Due to the unrealizable nature of this interpretation, Us no longer exists explicitly. Also, the saturation block is replaced by a dead zone block cp, where: u = cp(u) = U - 4>(u) (2)

This paper uses LC to deal with input saturation and joint torque limitation issues in a motion control setting. The saturation limit of the Le compensator depends on both actual and virtual limits. The use of LC in this manner makes it possible to utilize the actuator and HD closer to their respective limits. The a posteriori nature of the Le used in this paper means that it can be used to complement an existing controller that do not explicitly consider saturation.

The intended linear system captures the behavior of the "ideal" system, where its behavior depends only on G and K . The nonlinear loop captures the nonlinear behavior of the nonlinearity as well as the choice of M. Finally, the disturbance filter determines how the" disturbance" ucaused by the nonlinear loop affects the system. Since N = CM, the choice of M affects both the nonlinear loop and the disturbance filter. Hence, a perfect cancellation of the undesirable effects of saturation may not be possible.

2. BASIC CONCEPT OF LC The structure of LC as described in (West on & Postlethwaite, 2000) is presented here to introduce the basic notation and concept of LC. Weston and Postlethwaite have shown that LC methods can be cast in the manner shown in Fig.1. K represents an existing controller that was designed without taking any saturation into account. Zd is the reference signal. The plant is represented by G and a memoryless, time-invariant input saturation 4> with a constant limit Uact > 0 where _ A..( ) _ { Uact sign (u) , lul ~ Uact (1) Us - '+' U I

u

..... _Jioear ~iliooiol: . . .

In addition to describing the Le structure shown in Fig. 1, Weston and Postlethwaite proposed the following Direct M Design (DMD). Let (A, E, C, D) be the minimal order state space realization of G. Then, a state space realization for M, with (3) lim M(jw) = I w_oo and without introducing additional state variables, is ([A + BFI, E, F, I). Hence, M - I and N are implemented as

, ul < Uact

Assume that the measured output y of the plant is the controlled output, i.e. y = z. LC controller design involves designing the M block, which can either be a constant or a dynamic system.

xic =

[A + BFI Xlc + Bu

[~;]=[C:DF]Xlc+[~]U

(4)

The Le gain F must be chosen such that [A+BF] is stable. it is the difference between the actual and saturated control effort. U c is the output of the M - I block, while Zc is the output of the

N is equal to C model M, where Cmodel is the model of the plant G. Le generates two outputs, namely U c and Zc. They modify the output and input of K, respectively.

428

N block. DMD essentially involves choosing F. One benefit of DMD is that the LC structure is independent of the original controller K already in use, unlike many LC methods. Then, the design of the original controller K and the additional LC are nearly decoupled. The design process involves the iteration of different values of the LC gain F based on the model of the plant, at the same time making sure that the stability of the nonlinear loop is preserved.

the HD gear ratio. k j and bj are the joint stiffness and damping constants. OJ is the joint deflection. Based on Eq. 5, the transform domain relation from Us to the HD torque Tk, can be written as Tk j

= GTj/usUs

(7)

where

The inverse of a square system (A, B, e, D) is ([A - [BD-le]], BD-I, -[D-1ej, D-l) (Misra & Patel, 1988). Hence, EqA assigns the plant poles as the zeros of M, regardless of the value of F. This makes sense from the point of view of the disturbance filter N, which could also be written as the cascade of G M. In steady-state, any resonance mode from the plant is cancelled by an exact anti-resonant mode of M . In other words, the disturbance filter N will be characterized by the zeros of the plant G and the poles of M. DMD can be interpreted as choosing the poles of M such that the disturbance filter attenuates disturbances as fast as possible, while ensuring stability of the non linear loop.

Suppose the HD has a maximum allowable torque T max, which may be associated with a particular HD service life, or artificially introduced to minimize the effects of HD nonlinearities. Consider Us in the extended L2 space (L 2e ), which is arbitrarily long but finite in duration. Given Eq.7 and T rnax, a virtual saturation limit Uvir can be computed by defining 'hj / u.

:=

mJX IGTj / tts (jw) I

U'vi,·

:= T max/"ITj / u S

(9)

where "ITuu , is the gain from Us to Tk j , for Us E L 2 e. (Astrom & Wittenmark, 1995). Hence, for Us E L 2e , Iusl Uvir -+ ITkj Trnaxl· Define

3. LC FOR A SINGLE-JOINT ID WITH ACTUATOR AND HD TORQUE SATURATION

:s

Umin

:s

:=

min(uaet, Uvir)

(10)

Then, the actual constraints to the system, Us and T rnax , are replaced by a single bound Umin' Ensuring 11.11 Umin means that both the actuator saturation and the virtual saturation (due to HD torque limitation) will not be exceeded. This basic idea can be used to maximize the utilization of both the HD and actuator from control design and physical design points of view.

:s

Fig. 3. Plant model Consider a single-joint ID unit with a HD (Fig.3). The HD is modelled as a rigid gear plus a linear spring. While the HD possesses several nonlinearities, experimental results from (Taghirad & Belanger, 1998) and (Hashimoto & Kiyosawa, 1998) for example, have shown that a linear spring model is sufficient for feedback. The motor and load-side dynamics can be written as .. . 1 [ .] JmBm = ktu s - bmBm + N kjoj + bjoj

3.1 Control Design Point of View

(6)

For a particular choice of HD, Urnin can be used as the conservative input saturation value of the LC. This means that the controller is designed with a limit Umin such that neither the actuator saturation limit nor the HD torque limit are exceeded at any time. Then, the LC structure can be directly implemented with Urnin as its saturation limit instead of Uaet .

The subscripts rn, j, and l correspond to the motor, joint, and load side. B corresponds to angular position. k t is the motor constant. N g is

However, if the joint is not well damped, the value of Uvir may be much lower than Uact . Hence, it makes sense to improve joint damping. Modify the system (see FigA) with

9

Jl81 = kjoj

+ bj 8j

(5)

where

Bm . Bm' OJ = - - -Bl' OJ = - - -Bl Ng

Ng

429

(11)

u

where Kv is added to improve damping, and is a function of some available measurement Yv' In Fig.4, 4> (with bounds at ±Uact) is the true actuator saturation, while cP (with bounds at ±Umitl) is a component of the Le. By improving damping (of a poorly damped system), the peak of the frequency response from U2 to Tk j can be made smaller than the undamped case (Eq.7) . As a result, 1TU U' is decreased, which increases Uvi,·'

Fig. 5. Uvir and

G/ U

G ,

-

N

TJ/.,,2 - f33s3

+ [cokt~J, + f321s2 + f31S + f30

(13)

0, 1, 2,3 as defined in Eq.8. The

k,ti ./' ,

-;;;:.:!

j

+ J; I -

C6 k ,kj

2N g J m

2

A

'

S Co

U"ir :=

. . d IS mcrease ,

~

J

(16)

corresponds to the virtual saturation input of a well-damped system.

As the parameter(s) of K" are varied to increase (, increases. However, since 1,, / u2 must be considered, Uvir needs to be compared with U act/1,, /,,2 instead of Uact. For simplicity, assume Kv only has 1 tunable parameter. This is illustrated in Fig.6. One possible scenario is when Uvir < uact/1 u/ .. 2 , 'V( ::5 1 (Fig.6a) , and arIother is when Uvir = uacthuM for some ( :::; 1 (Fig.6b). In the 1st case, it makes sense to choose Kv that maximizes Uvir , while in the 2 nd case, it makes sense to choose Kv such that Uvir = u ac t/'U / U2' Then, the utilization of the HD and actuator are maximized. Uvir

Having determined the parameters of K v , instead of using Eq.lO to determine Umin for the Le, use

p J

Tma xl1T - J'/ U 8

Uvir

the real component of the poles becomes more negative while the magnitude of the imaginary component decreases. When Co is further in1 qk,kj 2 th creased such t h at k j [ N;J + J;1 I Uvin V ( :::; 1, and Fig.5b is when Uac! = Uvir at some ( :::; 1. Eq.12 also improves joint damping even when the HD stiffness behaves like a hardening spring with the form Tk = ' " ki o[,2 i-l ]

:=

In addition, given a particular joint damping controller K" , there exist a -1rJ'/ U 8 > 0 such that U"ir cannot be made any larger without increasing Tma:r.. Let

poles of G Tj / u2 in Eq.13 and GTj / UB in Eq.7 due to Eq.12. When the differs by Cn 9 joint is poorly damped, bm , bj ~ O. Then, the poles of GTj / u 2 (Eq.13) are at ± 1

u2

Recall that Eq.12 was introduced to reduce 1Ti/U> , thereby increasing U vir in systems with poor damping. However, if Eq.12 cannot be implemented , and either an approximation to the derivative or another vibration attenuation controller is to be used in Eq.l1 , it is possible that 'V > 1. Then, choosing Urn'i n according to 'u / u2 Eq.l0 does not guarantee lul :::; Uact . Hence, if improving system damping is achieved such that "V ',, /11. 2 > 1, then the increase in Uvir is achieved at the expense of lowering the limit due to actuator saturation by a factor of 1 u / u 2'

Fig. 4. System with Le and joint torque derivative

1 k j [ N;J m

+ f32s2 + f31S + /30 /33 s3 + [co kt~jJI + /321s2 + /31 S + /30 9 m;x IG'f! u2 (jw) I (15) , ~ \ .

=--.:....=..---,-~:;:.-------

It can be shown that uslll~, Eq.12we obtam 1u / u2 = 1, VCo 2: O. This unity gain allows the joint torque derivative feedback to be placed between if; and 4> in Fig.4 without affecting the Le structure.

9

with f3i , i

as joint damping is increased

'

= GTj / ,,2 U 2 g

1~

f33s3

1 ,, / ,,2

with Co > O. As long as lul :::; Usat , the transform domain relation from the new control input U2 to Tk j is -~JIS

Uact

(b)

In the linear case, the transform domain relation from U2 to U and its gain 1 u / u 2 are

Increasing Uv ir by improving joint damping can be done in many ways. For example, if Tk j is available, define (12)

Tk j

1~

(a)

(14)

i=1

Umin :=

where k i > 0, i = 1, .. . ,p (Budiman, 2001).

min(_l- uact ,uv ir) 'U/,,2

430

(17)

Lt

uac •

Table 1. Parameter values

Lt

Uacl

Yu/u2

Parameter 11 Value

(a)

Fig. 6. 'Uvi,· and

I ~

'UacL!"'t u / u ,

1.00 1.50 4.00 4.00 3.20 3.15 80

Jm Jl kj bj bm kt

U vir

(b)

as a function of (

Ng

3.2 Physical Design Point of View

x x x x

10-' 10+ 1 10+ 3 10+ 1 X 10- 2 x 10 - 1

[Units] [Nms< /rad] [Nms2/rad) [Nm/rad) [Nms/rad) [Nms/rad) [Nm/A]

reference. The system has an existing Kz which do not take actuator saturation nor HD torque limit into account. K = -37 [.258 + 1] ( ) z [.006258 + 1][.0058 + 1] 19

The physical design of a robot is subject to many constraints and considerations, which may be inter dependent. Even when the HD type is the only choice left to determine, there are still many factors to consider.

The tracking performance to a step reference

Suppose during the physical design process, there are several candidate HD with the same gear ratio N g for a particular robot design. Based on the previous discussion on LC, an additional guideline that will prevent under-utilization of both the actuator and HD can be considered. During the physical design stage, many of the physical parameters such as bj and bm may not be well known. In addition, any joint vibration damping scheme (which has not been determined yet during the physical design process) may result in "'tu / u , ~ 1. Recall that joint vibration damping results in an increased value of UviT> but scales Usat down by a factor of "'tu / u ' (see Eq.17). Since the value of "'tu / u ' is not known in advance unless joint torque derivative h; available for feedback, it makes sense to choose the HD to be on the weak side. In other words, assuming the HD is the only component left to be determined, choose one among the candidates with a gear ratio N g such that (18) Uvir ::; 'Uact

Fig. 7. Measured response to a step reference when no Input constraint is imposed is shown in Fig.7. This figure shows data from four different experiments. From top to bottom, the plots show the link position, control effort, motor position, and HD torque. Note that both the control effort and HD torque in this case are smaller than their respective limits of 12 A and 137 Nm. The variation in response among the four data sets are due to nonlinearities that are not considered in the modelling and control design.

4. EXPERIMENTAL RESULTS

Taking the Coulomb friction compensation into account, 'Uact = 11.65 A. With the joint torque sensor, an approximate derivative of Tkj is used to increase joint damping:

An existing single-joint ID unit is used for experimental verification. A FANUC a1 \3000 brushless DC motor is connected to the joint via an HD Systems CSF-25-80-2UH harmonic drive. A FANUC A06B-6089-H101 servo amplifier unit drives the motor. A FANUC A860-0316-TOOl HR pulse coder measures the motor's angular position. A Minebea TMNR-50KM torque sensor measures HD torque. Coulomb friction on the motor side is compensated using a method outlined in (Budiman & Tomizuka, 2002). System identification is then performed on the Coulomb friction compensated system. For this paper, the plant is assumed to be linear and is modelled by Eq.5. The values are shown in Table 1. The control objective is to make (J/ follow a particular

(fa

'Ul

= KvTk ,· = c/j--T k 8 + (fa '

(20)

with Co = .00095 and (fa = 157 rad/8. (fa is limited by the hardware, which in turn limits C/j. This results in the largest increase in joint damping, with "'tu / u , approximately equal to 2, and 'Uvir = 4.2 A. Using Eq.17, 'Umin = 4.2 A . For the experiment, an additional safety factor of 2 is used. Hence, 'Umin is considered to be equal to 2.1 A. To show the performance degradation when saturation becomes an issue, this artificial limit of 2.1 A is imposed on the system.

431

~:~~: : : : : : : 1 ':l~~ ~.~ ~:~! ~,.: j

be shown that joint damping can be increased. This results in a better utilization of the HD. If other vibration damping methods are used, and the resulting 1,, / .. 2 is larger than 1, then the increase in U lI ir is achieved at the expense of having to adjust the actuator saturation limit Uact by a factor of ',, / u2' In practical terms, this means that the vibration damping should be adjusted such that Uvir = ~. The effect of lu / u2 LC in a single-joint ID unit was demonstrated experimentally. In addition, this approach can be used to add saturation considerations in the physical design of robots.

. ..

!-

; ~:t:s;:d: ';;" ~' .~ "] ~~~\:; '~~" >;:. J o

0.2

o

0.2

0.4

0.'

0 .'

0.'

01

...;. 0 .•

1

U

H

"

I'

1..

1.'

¥ .;•. ,.;", 1

1.2

t..C}

U

Fig. 8. Measured response to a step reference when an artificial saturation limit of 2.1 A is imposed

ACKti°WLEDGEMENT

. research was' funped \. . ThIS by a grant prOVIded by FANUC Ltd., Japan. ' 6. REFERENCES

Fig.8 shows the performance degradation of the system in the presence of a 2.1 A saturation limit. To reduce the performance degradation due to saturation shown in Figure 8, an LC compensator is designed using the Direct M Design method. After several trial and error, the LC feedback gain is set to F = [-645.2 - 449.6 - 143.7 - 21.2]. Results from four different runs are shown in Figure 9. Compared to Figure 8, Figure 9 exhibits performance much closer to the unsaturated case, even with the ±2.1 A limit imposed.

!~:~tZ: o

0 .2

0 .4

'

o.e

, 0 .8

' .2

...:

, 1.1

Astrom & Wittenmark (1995) . Adaptive Control. Chap. 2, pp. 216-217. 2 ed .. Addison-Wesley. Budiman (2001). Torque Feedback & Saturation Compensation for Motion Control of an Indirect-Drive Unit using a Harmonic Drive. PhD thesis. U. of California at Berkeley. Budiman & Tomizuka (2002) . Lyapunov-Based Tuning Guideline of Motor Velocity Estimate for Coulomb Friction Compensation in a Flexible Joint Robot. In: Japan- USA Symp . on Flex. Automation. Hiroshima, Japan. Campo & Morari (1990). Robust Control of Processe::; Subject to Saturation Nonlinearities. Comp'uters fj Chem. Eng. 14, 343-358, Hanus, Kinneart, & Henrotte (1987) . Conditioning Technique, a General Anti-Windup & Bumpless Tran::;fer Method. Automatica 23, 729-739, Hashimoto & Kiyosawa (1998) . Experimental Study on Torque Control Using Harmonic Drive Built-in Torque Sensors. J. of Robotic Systems 15, 435-445, Hippe & Wurmthaler (1999), Systematic ClosedLoop Design in the Presence of Input Saturations, Automatica 35, 689-695. Kothare, Campo, Morari, & Nett (1994) , A Unified Framework for the Study of Anti-windup Designs. Automatica 30, 1869-1883, Misra & Patel (1988) , Transmission Zero Assignment in Linear Multivariable Systems Part I: Square Systems. In: Proc. of IEEE CDC. Austin, Texas. pp. 1310-1311. Taghirad & Belanger (1998) . Modelling & Parameter Identification of Harmonic Drive Systems. J. of Dyn. Sys ., Meas., fj Control 120, 439-444. Weston & Postlethwaite (2000) . Linear Conditioning for Systems Containing Saturating Actuators. Automatica 36, 1347-1354.

i 2

Fig. 9. Measured response to a step reference when LC is used, and the 2.1 A artificial saturation limit is imposed

5. CONCLUSIONS LC is used to maximize the utilization of the actuator and HD of an ID unit. By computing a virtual input Uvir which causes the HD torque to equal its allowable limit T max , it is possible to consider both actuator saturation and HD torque limit in the controller design. The value of Uvir can be made higher in poorly damped systems by adding a joint vibration attenuation controller. If joint torque derivative feedback is possible, it can

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