Underactuated Manipulator Robot Control By State Feedback Linearization Via H∞

Copyright a> IF AC Robust Control Design. Prague. Czech Republic. 2000

UNDERACTUATED MANIPULATOR ROB(}T CONTROL BY STATE FEEDBACK LINEARlZA TION VIA H..

Marco Henrique Terra, Benedito C. O. Madel, Paulo H. R. Nakashima*, Marcel Bergerman **

* Electrical Engineering Depanment - EESc/uSP P.O. Box 359, 13560-970, Siio Carlos, SP, Brazil e-mails: [email protected]

cn

** Automation Institute P.O. Box 6162, 13083-970, Campinas, SP, Brazil

Abstract: In this article we address the important problem of robust control design for an underactuated manipulator robot with three joints. The H-infinity design procedure considers the generalized regulator problem, and a suboptimal H-infinity controller is designed. This control strategy is compared with a known PID control for different structures of the underactuated robot. Tests are performed to verify the controller's robustness. Copyright C 2000 IFAC Keywords: Underactuated robotic manipulators, H-infinity control, Robustness, Feedback linearization, Passive joints.

1. INTRODUCTION

linearizatioo plus PID controller. The simulations are performed in our underactuated manipulatCl' control system development envirooment. reported in (Terra et al., 1999).

In this article we address robust control design for an underactuated manipulator robot with three joints. Underactuated manipulators are mechanisms composed of both active and passive (unactuated) joints, which arise in practice either due to joint failures, or as a result of robot design. Our contribution resides in the applicatioo of H.. control to position all joints of an Wideractuated manipulator. The controller is designed considering the generalized regulator problem to the configuratioos of the underactuated system.

2. UNDERACTUATED MANIPULATORS

In this section we briefly summarize the main concepts of underactuated manipulator theory available in the literature (Arai and Tachi, 1991; Arai et al, 1993; Bergerman, 1996). Underactuated manipulators differ from fully actuated ooes in that the formers are equipped with a number of actuators that are always smaller than the number of degrees-of-freedom (DOFs). Therefore, not all DOFs can be actively controlled concurrently. The vector of generalized coordinates is divided into two subsets, respectively representing the active generalized coordinates (i.e., the ooes being controlled at any given instant) and passive generalized coordinates (i.e., the remaining ones). Consider an n-linIc, open chain, and underactuated manipulator with rigid links. Let q represent its joint vector and 't represent its torque vector. The dynamic equatioos of the manipulator are found in closed-form via classical Lagrangian approach as:

This article is structured as follows: In section 2 we present the three control strategies available for underactuated robot manipulator control. In the first strategy, only the positions of the active joints are controlled. When this is the case, all other joints (i.e., the passive joints) are kept locked in place. This choice allows us to control the active joints as if the manipulator were fully actuated. In the second strategy, only the positions of the passive joints are taken into account (when the number of passive joints is at least equal to the number of active joints). In the third strategy, the vector of the controlled joints may contain both active and passive joints.

1'= M(q)i:j+~q, q)

The H.. controller design procedure is described in section 3. In section 4 the performance of this robust controller is compared with that of Arai and Tachi (1991), who proposed a feedback

(1)

In equation (1), M is the n x n symmetric, positivedefinite inertia matrix. Coriolis and centrifugal

51 3

tenns, gravitational torques, and frictional torques are grouped in the vector of non-inertial torques b. n. degrees of freedom of the manipulator are active joints with actuators and displacement sensors, which is a typical structure of manipulators joints. The remaining np = n - no) degrees of freedom are passive joints that have holding brakes instead of actuators.

disturbances, substitution of u in (3) by a PID-like controller of the form

There are three possible ways of forming the n. vector of active generalized coordinates qc (vector composed by the joints that are being controlled), each one leading to a different control strategy for the underactuated manipulator. The first case can be referred to as control strategy A, because only active joints are controlled. The second case can be referred to as control strategy P, because only passive joints are controlled. Finally, the third case is named an AP control strategy, because both active and passive joints are controlled.

In this article we propose to substitute the PID-type control law by one designed on top of H.. control theory, as described in the sequence. The block diagram of the proposed controller can be seen in Figure 1.

With this categorization of the possible control strategies for underactuated manipulators, a sequential control method to bring all joints to their set-points can be described as follows : first, a P or AP strategy is executed so that all passive joints (and perhaps some active joints, when n. > np) are controlled to their desired set-points. Then an A strategy is executed to bring the remaining active joints to their desired position. In other words, underactuated manipulators with more active than passive joints can be controlled to an equilibrium point in two phases, P-A or AP-A, provided, of course, that the passive joints are controllable via their dynamic coupling with the active ones.

Figure 1. Block diagram of the proposed feedback linearization plus robust controller.

guarantees that qc follows a desired trajectory q~ (I) .

3. H.. CONTROL DESIGN

H.. is a controller design method based on optimization, which guarantees stability and controlled system performance robustness. The purpose of this section is describe the design procedure of an H.. controller (Chiang and Safonov, 1992; Doyle el ai. , 1989; Francis, 1987; Maciejowski, 1989; Safonov et aI., 1987, 1989; Zhou el aI., 1995).

All three control strategies above lead to open-loop relationships between

qc

The H.. norm of a multivariable transfer function G is:

and t . (the torques

applied at the active joints) of the form :

!IGIL = supa we

(2)

QC

(6)

w.-·

be a high-pass filter that has the same gain Let characteristics of the sensitivity function S (defined as S = [J + G(s }K(s ):tl) at low frequencies. Then

(7)

(3)

M

(G(jm»

where sup is the supremum and O"max denotes the maximum singular value. The H.. nonn can then be used to "shape" or to "restrict" transfer functions .

The definitions of M ac and ba for strategies A , P. and AP can be found in Bergennan (1996). The resemblance of this dynamic equation with that of a fully actuated manipulator led Arai and Tachi (1991), to choose a feedback linearization controller to drive the joints in qc. The method consist in defining an auxiliary input u with:

such that, when the inverse of

Mall

9\

It is desired to restrict the maximum peak of the sensitivity function to avoid excessive disturbance amplification. On the H.. control formulation this can be made explicitly. Choosing an adequate weight function WJ , it is possible to "shape" the complementary sensitivity function C, defined as C = [I + C(s )K(s )tIG(s )K(s) . Therefore

exists,

(4) The effect of the linearization controller (3) is to uncouple and linearize the nonlinear system (2). In the absence of modeling errors and external

514

(8)

(13)

Combining (13) with (9) results in 3.1 The generalized regulator problem

Many control problems can be viewed as H.. control problems. In practice, one wishes to find a controller that satisfies simultaneously some objectives, such as disturbance rejection and sensor noise attenuation. This requires that the inequalities (7) and (8) be satisfied for the same controller K. Consider a linear system, described by the transfer function P (see figure 2).

z

The transfer function matrix (14) occurs frequently in H.. control theory, and it is called a linear fraction transformation (LFr) of P and K, denoted g(p, K), i.e.:

w

The generalized regulator problem may be formulated as follows. Given P, find K such that I.

g(p, K)

2.

Ilg(p, K~L ~ 1

is internally stable;

Figure 2. The generalized regulator. 3.2. The H_control solution

System P has two sets of input signals, wand u. The vector u contains all control inputs which can be manipulated in such a way that the objectives are reached. The vector w contains all other inputs, such as reference signals, disturbances, sensor noise, etc. System P has two sets of output signals, z and y. The vector y contains all measured variables. The vector z contains all other variables that one wishes to control. Mathematically we have:

The solution described here is a result obtained by Doyle et al. (1989). This solution makes use of the resolution of two algebraic Riccati equations (ARE) and designs a controller with the same number of states of the plant. Given a block system P 2x2 and an upper limit 'Y to the closed loop gain, the solution returns a compensator K(s) that stabilizes the system such

thatll~(P,

The result that follows is based on certain constraints on matrix D of the transfer function matrix of the plant P (the matrix D of P is the matrix formed by Du, D 12 , D2/, D22 , which appears in equation (16». These constraints may be ignored when applying the results of Safonov et al. (1989). The realization of the transfer matrix P is of the form

The generalized regulator problem consists of designing a linear system K such that, with u K y, the closed loop system (figure 2) from w to z, has the following properties:

=

1.

The closed loop is internally stable;

2.

The H.. norm of the closed loop is less than or equal to I.

Kt S; r .

(16) The closed loop transfer function matrix is computed as follows:

u=Ky

which is represented by the following set of equations:

(10) (11)

(17)

Therefore:

y

= [(1- P22 K )-1 P21 )v

(18) (12)

(19)

and from (10):

The following assumptions are made:

515

(i)

(A, Bl ) and (A, B2 ) are stabilizable;

(ii)

(CJ, A) and (Cl, A) are detectable;

The H.. solution involves two matrices, given by (20) and (21):

H '•.-

4. CONTROL OF UNDERACfUATED MANIPULATORS VIA H_ CONTROL In this section we apply the H.. control design presented above for the position control of an underactuated manipulator. In other words, we are interested in bringing all joints to their set-point regardless of them being actuated or not. The manipulator's inertia matrix M and inertial torque vector b utilized are shown in Terra et al (1999), corresponding to the actual parameters of a three-link manipulator built at Camegie Mellon University, USA. These parameter are shown in table 1.

Hamiltonian

A [-C'C , ,

y-'B,B: -B,B;]

A'

-'C'C -C'C ]

(20)

-A'

Table 1 Robot parameters mi (kg)

J := • [ -B,B;

y"

(21)

2'

-A

2 3

Theorem: There exists an admissible controller such that

liTzw IL < r

three conditions hold: H.. E dom(Ric) and X_ := Ric(H..)

~

(ii)

J_ E dom(Ric) and y_:= Ric(J_)

0;

~

0.0153 0.010 0.010

0,Q3 0,06 0,06

0,203 0,203 0,203

The linearization was realized considering seven possible configurations, combining active (A) and passive (P) joints. The possible configurations are: AAA, AAP, APA, PAA, APP, PAP, and PPA. Following the H.. design procedure, controllers given by (22) were calculated for three configurations, AAA, AAP and APP, and then their static portions (F-) were tested in our simulation environment (Terra et aI. , 1999).

if and only if the following

(i)

L;(m)

1,83 0,81 0,81

1

0;

(iii)

To obtain the space state matrices A, B, C, and D for the design of the H.. controller for the AAA configuration, the following method was used.

where p(XJJ=m:u:IA,(XJ.)1 is the spectral radius of X_Y_, Ai(X ].) is the i-th eigenvalue of X_Y_, and the notation X = Ric (H) is used to denote a solution for the ARE that makes E-RX stable (all of its eigenvalues are on the left half plane).

The states are defined as Xl = qI, x2 x3 = q2 ' x6

The process of finding a suboptimal H.. controller is iterative. Beginning with an arbitrary value of y, e.g., y = I, if one of the conditions above fails, then y is too small and a solution does not exist; therefore, y should be increased.

x4

=Xl =Ih '

Xs = q3 '

and

= Xs =43 z

The signals z2

= x3 = 42 ,

=Xl =ql ' z3 = Xs = q3 '

and y are defined as zl

= x3 = q2 ,

Moreover, when these conditions hold, one such con troller is Therefore, the state matrices are: (22)

where

o 100 0 0 000000 000100 A= 0 0 0 0 0 0 B - [B,

A. := A + r -'B,B: X. + B,F. + Z.L.C"

F. :=-B;X., L. := -Y.C; and Z. := (l-r-'Y.X.

000001 000000

t

I

This suboptimal H.. controller is often called the central controller or minimum entropy controller.

0 0 000

1 o 0 0 o 0 0 0 1 o 1 o 0 0 0 0 o 0 1 000 0 0

c=[~]=

516

o o

0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 000000000 B,)= 000 1 000 1 0

o

o

0 0 0 000 0 0 0 000 100 1

000 0 0 0 000 o0 0 0 0 0 0 0 0 o0 0 0 0 0 0 0 0 Du - 1 000 0 0 0 0 0 o 0 100 0 0 0 0 000 0 100 0 0

D=[I)o, DOl D"j _

torques applied on the active joints are considered as inputs of the system for each control stage.

. -' " -:; ;--: ::: ii;:t'

: r:-=~'­

·:f/·. -::. :. :::. .~ -.'. ,: I,, · . ,. '. : . ' . '. .' . ' . . - - . - . ..

.

.. . .

"l-.-:0'"7-;'0"-_;-,.-:..:--:---t.'~

"

(a)

(b)

Figure 3. Control of a three-link manipulator, configuration AAA. (a) Computed torque control. (b) H..robust control.

(a)

Figure 4. Control of a three-link manipulator, configuration AAP. (a) PlO control. (b) H.. robust control.

This will result in an H.. controller on the form: K =

0]

1.0819 1.5915 0 0 oo 0 0 1.0819 1.5915 0 [ o 0 0 0 1.0819 1.5915

(23)

Figure 4 compares the performance of the H.. controller (Figure 3b) with that of Arai and Tachi's PID-type control. The initial position is q(to) =[0°; 0°; 0°] and the desired position is qd = [30°; 30°; 30°].

Figure 3 compares the performance of the H.. controller (Figure 3b) with that of a standard computed torque one. The initial position is q(to) = [0°; 0°; 0°] and the desired position is qd = [30°; 30°; 30°].

For both controllers the graphics in Figure 4 clearly show the two stages of control: passive joint control followed by control of the active joints with the passive joint locked.

It can be seen that the joint angles converge

exponentially to the final desired positions. Compared to the computed torque, the H.. controller presents a slower position response, with differences on the convergence of the joint positions.

For the APP configuration, three steps are necessary for position control. In the first step, joint 2 (passive) is locked and joint 3 (passive) is controlled via its dynamic coupling with joint 1 (active). As explained in the previous item, when there are locked joints, the states related to these joints are cancelled in the state space matrices. In this case, the states that will be eliminated are X3 =q2 and x 4 =U 2' The obtained gain matrix (K1'3) will be identical to the KA,1' matrix above, but the last two columns are also cancelled, because 1:3 =0 (passive joint). In the second step the reasoning is the same, except that the cancelled states are those related to joint 3 (X5 and X6), which is locked in its final position. In the last step, joint 1 (active) is the only one being controlled, since joints 2 and 3 are locked. Therefore, the columns related to these joints (states X], ~ , X5 and Xis) will be eliminated, resulting in a 2x2 system, which leads to the same gain matrix of the first two steps. Therefore, the matrices are:

For configuration AAP, two control stages are necessary to control all joints. First, an AP strategy is applied where joints 2 and 3 are controlled. The computation of the gain matrix of this step (KA,I') is exactly equal to the computation of the AAA configuration, but the columns of the gain matrix which refer to the passive joint (columns 5 and 6) are eliminated, because 1: p =O. After the passive joint converges, strategy A is applied to control the active joints 1 and 2 to the desired position. In this second step, joint 3 is locked and joints 1 and 2 (active) are controlled. Here, X5 = q3 =0 and X6 =u3 =0 (joint 3 locked). This will result in a lower order system, because the rows and columns of X5 and X6 of the state space matrices will be cancelled. The computation of the H.. gain for this system (KA,), results in a gain identical to KA,I" Thus, the gain matrices for the two steps are:

KA =K AI,

=

[

10819 1.5915 0 0] '0 0 1.0819 1.5915 o 0 0 0

(b)

(24)

Figure 5 compares the performance of the H.. controller (Figure 3b) with that of Arai and Tachi's PlO-type control. The initial position is q(to) = [0°; 0°; 0°] and the desired position is qd = [30°; 30°; 30°].

The dimensions of the gain matrices (24) are smaller than in (23) because the number of columns of the matrix B is equal to the number of controlled joints na , which in turn is equal to the number of the active joints (Arai and Tachi, 1991). Only the

517

- -Ti-' .. , :/.""-"---"' I r :! ,: ,: :-::: c::

,/

.

-'

-'

"

'r-:-:'

~ -' -'/-' -'-'-' >-~\,-:- i - : -: ' :-

I . : : '-: -:

\" . -.

-. u

and good performance in the presence of parametric uncertainties.

T:

: !~::.~<: :

L -~~:::::!: ~

I ••

'

____ _

7. ACKNOWLEDGMENTS

:

•

We would like to express our gratitude to FAPESP for the financial support under grant nos. 9911 0031I and 97/13384-7.

u

(a)

(b)

Figure 5. Control of a three-link manipulator, configuration APP. (a) PID control. (b) I-L robust control.

8. REFERENCES Arai, H., Tachi, S. (1991). Position control of a manipulator with passive joints using dynamic coupling. IEEE Transactions on Robotics and Automation, 7, 528-534. Arai, H., Tanie, K, Tachi, S. (1993). Dynamic control of a manipulator with passive joints in operation space. IEEE Transactions on Robotics and Automation, 9, 85-93. Bergerman, M. (1996). Dynamics and control of underactuated manipulators. Ph.D. Thesis. Camegie Mellon University, Pittsburgh, PA, USA Chiang, RY., Safonov, M.G. (1992). Robust control toolbox. The Math Works. New York Doyle, le., Francis, B.A., Tannenbaum A.R (1992). Feedback control theory. Maxwell Macmillan. New York Doyle, lC., Glover, K, Khargonekar, P.P., Francis, B.A. (1989). State space solutions to standard H2 and H.. control problems. IEEE Transactions on Automatic Control, 34, 831847. Francis, B.A. (1987). A course in H_ control theory. Springer. New York. Maciejowski, lM. (1989). Multivariable feedback design. Addison-Wesley. Wokingham. Safonov, M.G., Jonckheere, E.A., Verma, M., Lebeer, DJ.N. (1987). Synthesis of positive real multi variable feedback systems. Intel7Ultional Journal of Control, 45, 817-842. Safonov, M.G., Limebeer, DJ.N., Chiang, RY. (1989). Simplifying the I-L theory via loopshifting, matrix-pencil and descriptor concepts. International Journal of Control, 50, 24672488. Terra, M.H., Siqueira, A.A.G., Bergerman, M. (1999). Underactuated Manipulator Robot Control via Linear Matrix Inequalities. IEEE Conference on Decision and Control-CDC. Phoenix, Arizona, USA. Thou, K , Doyle, lC., Glover, K (1995). Robust and Optimal Control. Prentice Hall. Englewood Cliffs, New Jersey.

5. ROBUSTNESS ANALYSIS Looking at the joint angles graphics in figures 3 to 5, one may come to the conclusion that the PID controller is better than the I-L, since the ' convergence time of the joints is smaller for the PID. One must note, however, an advantage of utilizing the I-L controller, which is the reason why it was proposed in this article: the robustness of the I-L controller under parametric uncertainties is far greater than that of the PID. To check this, the simulations were repeated for the AAA, AAP, and APP cases, with all robot parameters multiplied by four, i.e., an error of 300% in all parameters of the control model with respect to the simulation model. This error reflects possible disturbances in the inputs and outputs of the system, which is realistic assumption. Figures 6a and 6b present the results for the case AAA. One can see that the results obtained with the PID controller are worse than those obtained with the I-L controller. The same behavior has been verified for the other two cases.

/', . ,

(\l

,\ ~ <

if . "

>. -' .==;i .' . .

•

.

r:--' --'-

. i' .- - ~'<~-~ I~ '-

i\ >'-'.-:--:,-.' -.-: --- ' ~'

(b)

(a)

Figure 6. Robustness test for the underactuated manipulator robot, configuration AAA. (a) PID control. (b) I-Lrobust control.

6. CONCLUSIONS In this article, I-L control theory has been shown useful for robust control of all joints of an underactuated manipulator robot. The controller is designed considering the generalized regulator problem. The results obtained indicate some advantages of the proposed controller compared to a conventional PID controller, mainly its robustness

51 8