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Simple robust tracking controllers for robotic manipulators
Preprinll of !he Fourth IFAC Symposium on Robot Control
September 1~21. 1994. Capri.1taIy
SIMPLE ROBUST TRACKING CONTROLLERS FOR ROBOTIC MANIPULATORS SALAH ZENIEH
MARTIN CORLESS·
• School of Aeronautic6 and A6tronautic6, Purdue UniIJer6itll, Wed Lafallette, Indiana , USA
Ab.tract. We consider the problem of de.igning robust tracking controllers for uncertain rigid robotic manipulators. We propoee controllers which are robuot r - a tracking controllers in the following sense. For a prespecified rate of convergence a > and a prespecified tolerance r > 0, a proposed controller guarantees that the robot trajectory exponentially converges to any desired trajectory with rate a and to within the tolerance r. Controller design is based on Lyapunov functions . The main advantage of these controllers is their simplicity. Application to a two link robotic manipulator is presented. Numerical simulations are included .
°
Key Word •• Robust control . tracking control . robots , Lyapunov functions
1. INTRODUCTION
The two main approa.ches to designing robust traclting controller, for nonlinear robots are based on variable structure control (or "iding mode control) (Slotine and Sastry, 1983; Zinober, 1988) and Lyapunov functions (Corless, 1993). The latter approa.ch is taken in this paper.
The last decade has seen considerable research a.ctivity in the area of tra.cking controllers for rigid robot manipulators; see, e.g., (Ambrosino et al., 1988; Balestrino et al., 1983; Bayard and Wen, 1988; Corless, 1989; Ortega and Spong, 1989; Singh, 1985; Slotine and Li, 1987; Slotine and Sastry, 1983; Spong, 1992; Wen and Bayard , 1988; Zinober, 1988).
In the early a.ctivity on Lyapunov control design, the nonlinear robot is decomposed into a linear part and an uncertain/nonlinear part into which a.ll uncertainties and nonlinearities are lumped. Control design consists of three parts: a linear part to stabilize the linear system; a part to cancel out some portion of the nonlinearities; and a nonlinear part whose design is based on a. quadratic Lyapunov function and a norm bound on the remaining uncertain/non linear term (Corless, 1993).
The conceptua.lly simplest way to design controllers for rigid robots is the computed torque approach. This approa.ch is based on cancellation of a.ll system nonlinearities and, by redefinition of the control inputs, the system is transformed into a collection of decoupled double integrators. Because of the simplicity of a double integrator, controller design can readily be carried out to a.chieve many of the usual closed loop performance requirements. The main disadvantage of this approa.ch is that it requires exa.ct cancellation of system nonlinearities; hence the robustness of the approach is in question . By robustness of a controller we mean its ability to guarantee desired behavior/performance in the presence of uncertainties such as uncertain parameters, ina.curately modeled nonlinearities, and uncertain disturbance inputs. The other disavantage of this approach is the requirement of on-line computation in real time of the nonlinear terms which are being cancelled. These terms quickly become very complicated as the system order mcreases.
Recent controllers by Spong (Spong, 1992) are a.lso based on Lyapunov functions. In constructing the Lya.punov function, Spong takes advantage of the structure of the equations describing robotic manipulators and utilizes a certain skew-symmetry property which was popularized in the recent literature on adaptive control of robots; see (Ortega and Spong, 1989; Slotine and Li, 1987) . In Spong (Spong, 1992), the controller consists of a linear part, a part which cancels a nomina.! portion of the nonlinearities and a part whose construction is based on uncertainty bounds. By utilizing the skew-symmetry property and assuming that the system dynamics depend in a linear fashion on the uncertain
193
parameters, Spong obtains a control design for which computation of bounds is very simple; one just uses the bounds on the uncertain para.meters. However the implementation of these controllers require on-line computa.tion of a regressor matrix. This matrix depends on system dynamics and can quickly become very complicated as the system order increases. This paper also utilizes the above mentioned skewsymmetry property. The proposed controllers have a simpler structure than previous controllers based on Lyapunov functions and consist of a linear part and a nonlinear part. Controller design requires offline calculation of bounds associated with the nonlinear/uncertain term in the system dynamics. The computation of these bounds is simpler than the corresponding computations in earlier work on Lyapunov based controllers but, is not as simple as in Spong's results. However, since the bounds are computed off-line, this is not regarded as a major disadvantage. The main advantage of the controllers presented here is their simplicity ; hence they require relatively few on-line computations . Also the proposed approach does not make any assumptions on the manner in which the uncertainties enter the system description . The proposed controllers are r - a tracking controllers in the following sense. Given a prespecified rate of convergence a > 0 and a prespecified tolerance r > 0, a proposed controller guarantees that the robot trajectory exponentially converges to any desired trajectory with rate a and to within the tolerance r . The rest of this paper is organized as follows . We first present a precise problem statement. The next section presents the proposed controllers ; it also states the properties of the closed loop system resulting from these controllers. A procedure for calculating part of the uncertainty bounds is given next . A section describing Spong's robust controller follows. Finally we apply the proposed controller to a two link robotic manipulator . Numerical simulations compare the proposed controller to Spong's controller.
At time t E m., the vector q(t) E m." is the vector of generalized coordinates and u(t) E m." is the vector of control inputs. For a revolute joint robotic manipulator, the generalized coordinates are the angles which determine the locations of the links; the control inputs are usually torques applied to the links. The positivedefinite symmetric matrix M(q,6) is the system inertia matrix; the term G(q, cj, 6) represents all generalized forces (except those due to control inputs) acting on the system. Such forces could be due to gravity, friction , etc. The term C(q, q, 6)cj depends on the inertia matrix; by appropriate choice of C(q, cj, 6), the matrix
N(q,
q, 6)
:=
Consider the general class of uncertain mechanical systems described by the following equation:
IM(q, 6)q + C(q, cj , 6)cj + G(q, cj, 6) = I u
(1 )
194
(2)
is skew-symmetric (Ortega and Spong, 1989; Slotine and Li, 1987) ; also C(q, q, 6) depends linearly on q. The physical elements of inertia, mass, and any combination of these may be uncertain. All the uncertainty in the system is represented by the lumped uncertain term 6. The only knowledge that we have on 6 is that it belongs to a known nonempty set 6, i.e, 6 E 6 . Let qd(-) : m. -+ m. n be a prespecified, desired motion of the system. Ideally, we wish to design a controller for u such that every motion q( .) : m. -+ m. n of the resulting closed-loop system converges to the desired motion exponentially. Furthermore, we would like to specify the rate of convergence a priori. To be more specific, we will consider controllers with the following general form :
(3) This results in the closed loop system:
M(q, 6)ij + C(q, q, 6)q + G(q, q, 6) = p(q,q,qd , qd,qd)
(4)
Let a > 0 be the desired rate of exponential convergence and define the tracking error
q• := q - qd
2. PROBLEM STATEMENT
M(q, 6) - 2C(q, q, 6)
(5)
To require that the tracking error converges to zero usually requires a discontinuous controller which is undesirable for several reasons ; see (Corless, 1993). Hence, we relax our requirements and only require tracking to within some prespecified tolerance r . Specifically, given r , a > 0, we want to design a controller of the form (3) so that the resulting closed loop system (4) has the
where q = q - qd is the tracking error. Although it is not necessary, for simplicity of design we choose 11. to be a diagonal matrix.
following behavior.
Definition The clo$ed loop $Y$tem (,0 i$ an r-a robust tracker if there are scalar$ Cl and C2 $uch that, for any de$ired C 2 trajectory qd(-) and any uncertainty 6 E 1::., every $olution of thi$ $ystem $atisfies
3.1. Propo$ed Controllers The proposed controllers are given by
for all t
~
to.
(8)
In words, (6) means that the tracking error exponentially converges with rate a to the ball of radius r defined by B( r) := {q E m. n : Iq-I :5 r} For results on exponential convergence see (Corless, 1990). We now present an assumption that is a natural property of robotic manipulators.
where Q is any positive-definite, symmetric, matrix which satisfies
the scalars {31, {32, and {33 are chosen to assure the assumption; ( is any positive scalar satisfying
Assumption There exist (known) scalars {3o, {31, {32, and (33 $uch that, for all q, equalities hold I :
Amin[M(q, 6)] Amaz[M(q, 6)] IC(q, q, 6)1 IG(q, q, 6)1
> < :5 <
(3o
q,
and 6, the following in-
(9) and
>0
r
is the prespecified tracking tolerance.
{31 {321ql
3.2 . Closed Loop System Properties
{33
Our main result is summarized in the following theorem.
Theorem Consider any $Y$tem described by (1) which $atisfie$ the assumption and is subject to controller (8).
Section 4 contains a note on computing the bound {32 .
The resulting closed loop system is an r-a robust tracker with
3. PROPOSED CONTROLLERS AND THEIR
PROPERTIES In this section we present a control design procedure and state the properties of the resulting closed loop system. First choose any positive-definite symmetric matrix 11. which satisfies
(10)
Proof: For a complete proof see (Zenieh and Corless, n.d.). • where a is the desired convergence rate, and define the following variables: 4. COMPUTING THE BOUND {32 '1
In this section, we demonstrate how to compute the bound {32. First note that each component of the vector y := C(q, q, 6)/1 can be written as:
(7)
/I
I For a symmetric matrix M, A"" .. [Mj and >-", ... [Mj denote its minimum and the maximum eigenvalues , respectively.
195
where Li(q, 6) is a square matrix. Since the norm of y is given by
calculation of the bound pr is relatively easy. The approach we have presented in this paper eliminates the need for on-line calculation of the regressor matrix, hence, making controller implementation easier. The price we pay is in the calculation of the coefficients of the bounding function p(v, Ji, q). Although harder than computing pr, it is performed off-line and is done once.
we have
The required coefficient
/32
Remark Spong (Spong, 1992) proves that the a.bove controller achieves uniform ultimate boundedness (Corless and Leitmann, 1981) . One may readily show that this controller is also an r - ex tracking controller. This fact is also demonstrated in the simulations of the next section.
is given by :
V 6,q
6. EXAMPLE: TWO LINK PLANAR MANIPULATOR
5. SPONG'S ROBUST TRACKING CONTROLLER Spong's robust controller (Spong, 1992) for robotic manipulators requires an assumption popular in the adaptive control literature: linear parameterizability of the uncertain terms in the system dynamics. Specifically, the assumption is that
M(q, 6)';' + C(q,
q, 6)v + G(q, q, 6) = Y(q, q, v, ';')0(6)
where 0(6) is a vector containing all the uncertain elements of the system and Y(q, q, v,';') is termed the regressor matrix. The robust controller proposed by Spong is given by u
= -Q"., + Y(q, q, v, Ji)(8 + P.) 0
In this section, we apply the above two control designs to a two link robotic manipulator; see figure 1 for a simplified model. The coordinates q1 and q2 denote the angular location of the first and second links relative to the local vertical, respectively. The second link includes a payload of unknown mass located at its end. The masses of the first and the second links are m1 and m2, respectively. The moments of inertia of the first and the second links about their centers of mass are 11 and 12 , respectively. The locations of the center of mass of links one and two are determined by le1 and lC2, respectively; It is the length of link 1. There is an independent control torque applied to each arm, namely U1 and U2.
(11) Uncertainty
where yT
p.
=
-IYT~IPr {
-
yT!)
• pr
if
We suppose that the payload mass at the end of the second link is unknown. This is common in robotics where the controller is required to handle a variety of pay loads. Thus we consider the uncertainty to be 6 mpayload . Since m2, lC2 and 12 depend on the payload mass, these are uncertain parameters.
=
if
v and "., are as defined earlier; the vector 00 is a 'nominal' value of the uncertain vector 0(6) ; Q is a positivedefinite symmetric matrix; ( > 0; and the constant bound pr is chosen to satisfy 10(6) - 00 I ~ pr for all 6. Since this controller utilizes the regressor matrix, it is necessary to calculate this matrix on-line. The dynamics of robotic manipulators become increasingly complicated as the number of links increases which in turn makes this calculation time-consuming; however, the
196
Dynamics The equations of motion can be derived using Lagrange's equations. The system is described by (1) where the mass matrix M(q,6) is given by: mllc: [
+ m21i + 11
m 211 1c2C( ql - q2)
m2ltlc 2C(q1-q2) m21c~
+h
1
Payload
We allow the payload mass b to change within the following upper and lower limits : 0 :S b :S 5 . Then , for the second link, this range of b yields the following ranges for ffi2, iC2 , and h : 5 :S ffi2 :S 10
0.5:S
iC2
:S 0.525
5/12:S
h :S 20/12
Proposed Controller The details for the controller proposed in this paper are presented next. Given the above uncertainty, the bounds are computed to be: (30 = 0.83 , (31 = 18.21 , (32 = 10.6 1, (33 = 164 .35
The parameters we chose for the 1 , T = 0.3 Letting,
T-O'
tracker are:
0'
=
a proposed controller is given by (8) where ( is calculated using (9); we consid er ( = 0.07 Fig. 1. A Simplified Model of a Two link Manipulator
Spong Controller The following choice of C(q,q,b) guarantees the skewsymmetry of N(q, q, b):
Spong's choice of the uncertain parame ter vector B( b) IS
B(b) = The gravitational terms are given by the vector
ffidc~ + ffi2i~ + 11 ffi2ic~ + h ffi2i l /C2 ffil/CI ffi 2i l ffi2/C2
The transpose of the corresponding regressor matrix yT(q , q, v,v ) is given by: where 9 is the gravitational accele ration constant.
For simulation purposes, we will use th e same paramet er values as Spong (Spong, 1992) . The known paramete r values are presented in table 1.
Table 1
0.5
V2 C C>qVI - SC>qql VI
o wh e re
1
o
o CC>qV2 + SC>qq2 V2 -gSq, -gSq,
Physical Parameters
10
VI
q .) :=
o o -gSq,
cos( ·), So := sin(-) and t:J.q := ql - q2 .
The nominal value of B chosen by Spong for his simulations is given in table 2. Spong's controller is given by (11). We co nsider the same Q and A as above. Using the ranges of m 2, /C2, and h presented e arli er , the bound pr satisfies: pr :S 13.46 . The parameter ( is chosen to
10/12
Paramete rs of the unload ed arm
197
8.96
13.33
Table 2
8.75
5
10
nal of Dynamic Systems, Measurement, and Control 110, 215-220 . Balestrino, A., G. DeMaria and L. Schiavicco (1983) . An adaptive model following control for robotic manipulators . Journal of Dynamic Systems, Measurement, and Contrall05, 143-145 . Bayard, D.S . and J.T. Wen (1988). New class of control laws for robotic manipulators part 2. adaptive case . International Journal of Control 47, 1387-1406. Corless, M. (1989) . Tracking controllers for uncertain systems: Application to a manutec r3 robot. Journal of Dynamic Systems, Measurement, and Control 111, 609- 618 . Corless, M. (1990). Guaranteed rates of exponential convergence for uncertain systems. Journal of Optimization Theory and Applications 64, 471-484 . Corless, M. (1993) . Control of uncertain nonlinear systems. Journal of Dynamic Systems, Measurement, and Control 115, 362- 372. Corless, M. and G. Leitmann (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Transactions on Automatic Control AC-26, 1139- 1144. Ortega, R. and M.W. Spong (1989) . Adaptive motion control of rigid robots: A tutorial . A utomatica 25, 877-888. Singh, S.N . (1985). Adaptive model following control of nonlinear robotic. systems. IEEE Transactions on Automatic Control AC-30, 1099-1100. Slotine, J .E. and S.S . Sastry (1983) . Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators. International Journal of Control 48, 465-492 . Slotine, J .E. and W . Li (1987). On the adaptive control of robot manipulators. International Journal of Robotics Research 6, 49- 59. Spong, M.W. (1992). On the robust control ofrobot manipulators. IEEE Transactions on Automatic Control 37, 1782-1786. Wen, J .T. and D.S. Bayard (1988) . New class of control laws for robotic manipulators part 1. non-adaptive case. International Journal of Control 47, 1361-1385. Zenieh , Salah and Martin Corless (n .d .). Simple robust T - Cl' tracking controllers for robotic manipulators. Submitted for publication. Zinober, A.Z.I. (1988) . Variable structure modelfollowing control of a robot manipulator . In : 4th IFA C Symposium on Computer Aided Design in Control Syst ems. Beijing, PR China.
8.75
Nominal parameters
be the same as that for the proposed controller.
Numerical Simulations For simulation purposes, we chose m2 = 5.0, IC2 = 0.5, and h = i.e. the arm has no payload. In figure 2, the arm responses using both designs are overplotted along with the desired trajectory. The control history for both links is presented in figure 3.
A
1.1 ~
"0
f
u
~u
~ I. ' U
-, ~ , ---7"--7, ~~.~~-~
_U,L-----'--7, -~.~-----7-~
Time (sec)
Time (sec)
Fig. 2. Link Trajectories
,." ....--~:u..,Ul.L~.....",-r------, ~~{>'e!',~:::-::
S
-:''G
:i ." ...
~
.... ·1501
;:s ._
....
-
_.L._-'-----',. Time (sec)
__... Lo.._---'-_',
.1 111
. 11' L.o.._---'-_', _~.-----'--:'
Time (sec)
Fig. 3 . Control Torques
7. ACKNOWLEDGMENT The authors wish to acknowledge the support of the US National Science Foundation under Grant MSS-9057079 and Purdue University under a Purdue Research Foundation Grant .
8. REFERENCES Ambrosino, G., G. Celentano and F. Garofalo (1988) . Adaptive tracking control of industrial robots. Jour-
198
September 1~21. 1994. Capri.1taIy
SIMPLE ROBUST TRACKING CONTROLLERS FOR ROBOTIC MANIPULATORS SALAH ZENIEH
MARTIN CORLESS·
• School of Aeronautic6 and A6tronautic6, Purdue UniIJer6itll, Wed Lafallette, Indiana , USA
Ab.tract. We consider the problem of de.igning robust tracking controllers for uncertain rigid robotic manipulators. We propoee controllers which are robuot r - a tracking controllers in the following sense. For a prespecified rate of convergence a > and a prespecified tolerance r > 0, a proposed controller guarantees that the robot trajectory exponentially converges to any desired trajectory with rate a and to within the tolerance r. Controller design is based on Lyapunov functions . The main advantage of these controllers is their simplicity. Application to a two link robotic manipulator is presented. Numerical simulations are included .
°
Key Word •• Robust control . tracking control . robots , Lyapunov functions
1. INTRODUCTION
The two main approa.ches to designing robust traclting controller, for nonlinear robots are based on variable structure control (or "iding mode control) (Slotine and Sastry, 1983; Zinober, 1988) and Lyapunov functions (Corless, 1993). The latter approa.ch is taken in this paper.
The last decade has seen considerable research a.ctivity in the area of tra.cking controllers for rigid robot manipulators; see, e.g., (Ambrosino et al., 1988; Balestrino et al., 1983; Bayard and Wen, 1988; Corless, 1989; Ortega and Spong, 1989; Singh, 1985; Slotine and Li, 1987; Slotine and Sastry, 1983; Spong, 1992; Wen and Bayard , 1988; Zinober, 1988).
In the early a.ctivity on Lyapunov control design, the nonlinear robot is decomposed into a linear part and an uncertain/nonlinear part into which a.ll uncertainties and nonlinearities are lumped. Control design consists of three parts: a linear part to stabilize the linear system; a part to cancel out some portion of the nonlinearities; and a nonlinear part whose design is based on a. quadratic Lyapunov function and a norm bound on the remaining uncertain/non linear term (Corless, 1993).
The conceptua.lly simplest way to design controllers for rigid robots is the computed torque approach. This approa.ch is based on cancellation of a.ll system nonlinearities and, by redefinition of the control inputs, the system is transformed into a collection of decoupled double integrators. Because of the simplicity of a double integrator, controller design can readily be carried out to a.chieve many of the usual closed loop performance requirements. The main disadvantage of this approa.ch is that it requires exa.ct cancellation of system nonlinearities; hence the robustness of the approach is in question . By robustness of a controller we mean its ability to guarantee desired behavior/performance in the presence of uncertainties such as uncertain parameters, ina.curately modeled nonlinearities, and uncertain disturbance inputs. The other disavantage of this approach is the requirement of on-line computation in real time of the nonlinear terms which are being cancelled. These terms quickly become very complicated as the system order mcreases.
Recent controllers by Spong (Spong, 1992) are a.lso based on Lyapunov functions. In constructing the Lya.punov function, Spong takes advantage of the structure of the equations describing robotic manipulators and utilizes a certain skew-symmetry property which was popularized in the recent literature on adaptive control of robots; see (Ortega and Spong, 1989; Slotine and Li, 1987) . In Spong (Spong, 1992), the controller consists of a linear part, a part which cancels a nomina.! portion of the nonlinearities and a part whose construction is based on uncertainty bounds. By utilizing the skew-symmetry property and assuming that the system dynamics depend in a linear fashion on the uncertain
193
parameters, Spong obtains a control design for which computation of bounds is very simple; one just uses the bounds on the uncertain para.meters. However the implementation of these controllers require on-line computa.tion of a regressor matrix. This matrix depends on system dynamics and can quickly become very complicated as the system order increases. This paper also utilizes the above mentioned skewsymmetry property. The proposed controllers have a simpler structure than previous controllers based on Lyapunov functions and consist of a linear part and a nonlinear part. Controller design requires offline calculation of bounds associated with the nonlinear/uncertain term in the system dynamics. The computation of these bounds is simpler than the corresponding computations in earlier work on Lyapunov based controllers but, is not as simple as in Spong's results. However, since the bounds are computed off-line, this is not regarded as a major disadvantage. The main advantage of the controllers presented here is their simplicity ; hence they require relatively few on-line computations . Also the proposed approach does not make any assumptions on the manner in which the uncertainties enter the system description . The proposed controllers are r - a tracking controllers in the following sense. Given a prespecified rate of convergence a > 0 and a prespecified tolerance r > 0, a proposed controller guarantees that the robot trajectory exponentially converges to any desired trajectory with rate a and to within the tolerance r . The rest of this paper is organized as follows . We first present a precise problem statement. The next section presents the proposed controllers ; it also states the properties of the closed loop system resulting from these controllers. A procedure for calculating part of the uncertainty bounds is given next . A section describing Spong's robust controller follows. Finally we apply the proposed controller to a two link robotic manipulator . Numerical simulations compare the proposed controller to Spong's controller.
At time t E m., the vector q(t) E m." is the vector of generalized coordinates and u(t) E m." is the vector of control inputs. For a revolute joint robotic manipulator, the generalized coordinates are the angles which determine the locations of the links; the control inputs are usually torques applied to the links. The positivedefinite symmetric matrix M(q,6) is the system inertia matrix; the term G(q, cj, 6) represents all generalized forces (except those due to control inputs) acting on the system. Such forces could be due to gravity, friction , etc. The term C(q, q, 6)cj depends on the inertia matrix; by appropriate choice of C(q, cj, 6), the matrix
N(q,
q, 6)
:=
Consider the general class of uncertain mechanical systems described by the following equation:
IM(q, 6)q + C(q, cj , 6)cj + G(q, cj, 6) = I u
(1 )
194
(2)
is skew-symmetric (Ortega and Spong, 1989; Slotine and Li, 1987) ; also C(q, q, 6) depends linearly on q. The physical elements of inertia, mass, and any combination of these may be uncertain. All the uncertainty in the system is represented by the lumped uncertain term 6. The only knowledge that we have on 6 is that it belongs to a known nonempty set 6, i.e, 6 E 6 . Let qd(-) : m. -+ m. n be a prespecified, desired motion of the system. Ideally, we wish to design a controller for u such that every motion q( .) : m. -+ m. n of the resulting closed-loop system converges to the desired motion exponentially. Furthermore, we would like to specify the rate of convergence a priori. To be more specific, we will consider controllers with the following general form :
(3) This results in the closed loop system:
M(q, 6)ij + C(q, q, 6)q + G(q, q, 6) = p(q,q,qd , qd,qd)
(4)
Let a > 0 be the desired rate of exponential convergence and define the tracking error
q• := q - qd
2. PROBLEM STATEMENT
M(q, 6) - 2C(q, q, 6)
(5)
To require that the tracking error converges to zero usually requires a discontinuous controller which is undesirable for several reasons ; see (Corless, 1993). Hence, we relax our requirements and only require tracking to within some prespecified tolerance r . Specifically, given r , a > 0, we want to design a controller of the form (3) so that the resulting closed loop system (4) has the
where q = q - qd is the tracking error. Although it is not necessary, for simplicity of design we choose 11. to be a diagonal matrix.
following behavior.
Definition The clo$ed loop $Y$tem (,0 i$ an r-a robust tracker if there are scalar$ Cl and C2 $uch that, for any de$ired C 2 trajectory qd(-) and any uncertainty 6 E 1::., every $olution of thi$ $ystem $atisfies
3.1. Propo$ed Controllers The proposed controllers are given by
for all t
~
to.
(8)
In words, (6) means that the tracking error exponentially converges with rate a to the ball of radius r defined by B( r) := {q E m. n : Iq-I :5 r} For results on exponential convergence see (Corless, 1990). We now present an assumption that is a natural property of robotic manipulators.
where Q is any positive-definite, symmetric, matrix which satisfies
the scalars {31, {32, and {33 are chosen to assure the assumption; ( is any positive scalar satisfying
Assumption There exist (known) scalars {3o, {31, {32, and (33 $uch that, for all q, equalities hold I :
Amin[M(q, 6)] Amaz[M(q, 6)] IC(q, q, 6)1 IG(q, q, 6)1
> < :5 <
(3o
q,
and 6, the following in-
(9) and
>0
r
is the prespecified tracking tolerance.
{31 {321ql
3.2 . Closed Loop System Properties
{33
Our main result is summarized in the following theorem.
Theorem Consider any $Y$tem described by (1) which $atisfie$ the assumption and is subject to controller (8).
Section 4 contains a note on computing the bound {32 .
The resulting closed loop system is an r-a robust tracker with
3. PROPOSED CONTROLLERS AND THEIR
PROPERTIES In this section we present a control design procedure and state the properties of the resulting closed loop system. First choose any positive-definite symmetric matrix 11. which satisfies
(10)
Proof: For a complete proof see (Zenieh and Corless, n.d.). • where a is the desired convergence rate, and define the following variables: 4. COMPUTING THE BOUND {32 '1
In this section, we demonstrate how to compute the bound {32. First note that each component of the vector y := C(q, q, 6)/1 can be written as:
(7)
/I
I For a symmetric matrix M, A"" .. [Mj and >-", ... [Mj denote its minimum and the maximum eigenvalues , respectively.
195
where Li(q, 6) is a square matrix. Since the norm of y is given by
calculation of the bound pr is relatively easy. The approach we have presented in this paper eliminates the need for on-line calculation of the regressor matrix, hence, making controller implementation easier. The price we pay is in the calculation of the coefficients of the bounding function p(v, Ji, q). Although harder than computing pr, it is performed off-line and is done once.
we have
The required coefficient
/32
Remark Spong (Spong, 1992) proves that the a.bove controller achieves uniform ultimate boundedness (Corless and Leitmann, 1981) . One may readily show that this controller is also an r - ex tracking controller. This fact is also demonstrated in the simulations of the next section.
is given by :
V 6,q
6. EXAMPLE: TWO LINK PLANAR MANIPULATOR
5. SPONG'S ROBUST TRACKING CONTROLLER Spong's robust controller (Spong, 1992) for robotic manipulators requires an assumption popular in the adaptive control literature: linear parameterizability of the uncertain terms in the system dynamics. Specifically, the assumption is that
M(q, 6)';' + C(q,
q, 6)v + G(q, q, 6) = Y(q, q, v, ';')0(6)
where 0(6) is a vector containing all the uncertain elements of the system and Y(q, q, v,';') is termed the regressor matrix. The robust controller proposed by Spong is given by u
= -Q"., + Y(q, q, v, Ji)(8 + P.) 0
In this section, we apply the above two control designs to a two link robotic manipulator; see figure 1 for a simplified model. The coordinates q1 and q2 denote the angular location of the first and second links relative to the local vertical, respectively. The second link includes a payload of unknown mass located at its end. The masses of the first and the second links are m1 and m2, respectively. The moments of inertia of the first and the second links about their centers of mass are 11 and 12 , respectively. The locations of the center of mass of links one and two are determined by le1 and lC2, respectively; It is the length of link 1. There is an independent control torque applied to each arm, namely U1 and U2.
(11) Uncertainty
where yT
p.
=
-IYT~IPr {
-
yT!)
• pr
if
We suppose that the payload mass at the end of the second link is unknown. This is common in robotics where the controller is required to handle a variety of pay loads. Thus we consider the uncertainty to be 6 mpayload . Since m2, lC2 and 12 depend on the payload mass, these are uncertain parameters.
=
if
v and "., are as defined earlier; the vector 00 is a 'nominal' value of the uncertain vector 0(6) ; Q is a positivedefinite symmetric matrix; ( > 0; and the constant bound pr is chosen to satisfy 10(6) - 00 I ~ pr for all 6. Since this controller utilizes the regressor matrix, it is necessary to calculate this matrix on-line. The dynamics of robotic manipulators become increasingly complicated as the number of links increases which in turn makes this calculation time-consuming; however, the
196
Dynamics The equations of motion can be derived using Lagrange's equations. The system is described by (1) where the mass matrix M(q,6) is given by: mllc: [
+ m21i + 11
m 211 1c2C( ql - q2)
m2ltlc 2C(q1-q2) m21c~
+h
1
Payload
We allow the payload mass b to change within the following upper and lower limits : 0 :S b :S 5 . Then , for the second link, this range of b yields the following ranges for ffi2, iC2 , and h : 5 :S ffi2 :S 10
0.5:S
iC2
:S 0.525
5/12:S
h :S 20/12
Proposed Controller The details for the controller proposed in this paper are presented next. Given the above uncertainty, the bounds are computed to be: (30 = 0.83 , (31 = 18.21 , (32 = 10.6 1, (33 = 164 .35
The parameters we chose for the 1 , T = 0.3 Letting,
T-O'
tracker are:
0'
=
a proposed controller is given by (8) where ( is calculated using (9); we consid er ( = 0.07 Fig. 1. A Simplified Model of a Two link Manipulator
Spong Controller The following choice of C(q,q,b) guarantees the skewsymmetry of N(q, q, b):
Spong's choice of the uncertain parame ter vector B( b) IS
B(b) = The gravitational terms are given by the vector
ffidc~ + ffi2i~ + 11 ffi2ic~ + h ffi2i l /C2 ffil/CI ffi 2i l ffi2/C2
The transpose of the corresponding regressor matrix yT(q , q, v,v ) is given by: where 9 is the gravitational accele ration constant.
For simulation purposes, we will use th e same paramet er values as Spong (Spong, 1992) . The known paramete r values are presented in table 1.
Table 1
0.5
V2 C C>qVI - SC>qql VI
o wh e re
1
o
o CC>qV2 + SC>qq2 V2 -gSq, -gSq,
Physical Parameters
10
VI
q .) :=
o o -gSq,
cos( ·), So := sin(-) and t:J.q := ql - q2 .
The nominal value of B chosen by Spong for his simulations is given in table 2. Spong's controller is given by (11). We co nsider the same Q and A as above. Using the ranges of m 2, /C2, and h presented e arli er , the bound pr satisfies: pr :S 13.46 . The parameter ( is chosen to
10/12
Paramete rs of the unload ed arm
197
8.96
13.33
Table 2
8.75
5
10
nal of Dynamic Systems, Measurement, and Control 110, 215-220 . Balestrino, A., G. DeMaria and L. Schiavicco (1983) . An adaptive model following control for robotic manipulators . Journal of Dynamic Systems, Measurement, and Contrall05, 143-145 . Bayard, D.S . and J.T. Wen (1988). New class of control laws for robotic manipulators part 2. adaptive case . International Journal of Control 47, 1387-1406. Corless, M. (1989) . Tracking controllers for uncertain systems: Application to a manutec r3 robot. Journal of Dynamic Systems, Measurement, and Control 111, 609- 618 . Corless, M. (1990). Guaranteed rates of exponential convergence for uncertain systems. Journal of Optimization Theory and Applications 64, 471-484 . Corless, M. (1993) . Control of uncertain nonlinear systems. Journal of Dynamic Systems, Measurement, and Control 115, 362- 372. Corless, M. and G. Leitmann (1981). Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems. IEEE Transactions on Automatic Control AC-26, 1139- 1144. Ortega, R. and M.W. Spong (1989) . Adaptive motion control of rigid robots: A tutorial . A utomatica 25, 877-888. Singh, S.N . (1985). Adaptive model following control of nonlinear robotic. systems. IEEE Transactions on Automatic Control AC-30, 1099-1100. Slotine, J .E. and S.S . Sastry (1983) . Tracking control of nonlinear systems using sliding surfaces, with application to robot manipulators. International Journal of Control 48, 465-492 . Slotine, J .E. and W . Li (1987). On the adaptive control of robot manipulators. International Journal of Robotics Research 6, 49- 59. Spong, M.W. (1992). On the robust control ofrobot manipulators. IEEE Transactions on Automatic Control 37, 1782-1786. Wen, J .T. and D.S. Bayard (1988) . New class of control laws for robotic manipulators part 1. non-adaptive case. International Journal of Control 47, 1361-1385. Zenieh , Salah and Martin Corless (n .d .). Simple robust T - Cl' tracking controllers for robotic manipulators. Submitted for publication. Zinober, A.Z.I. (1988) . Variable structure modelfollowing control of a robot manipulator . In : 4th IFA C Symposium on Computer Aided Design in Control Syst ems. Beijing, PR China.
8.75
Nominal parameters
be the same as that for the proposed controller.
Numerical Simulations For simulation purposes, we chose m2 = 5.0, IC2 = 0.5, and h = i.e. the arm has no payload. In figure 2, the arm responses using both designs are overplotted along with the desired trajectory. The control history for both links is presented in figure 3.
A
1.1 ~
"0
f
u
~u
~ I. ' U
-, ~ , ---7"--7, ~~.~~-~
_U,L-----'--7, -~.~-----7-~
Time (sec)
Time (sec)
Fig. 2. Link Trajectories
,." ....--~:u..,Ul.L~.....",-r------, ~~{>'e!',~:::-::
S
-:''G
:i ." ...
~
.... ·1501
;:s ._
....
-
_.L._-'-----',. Time (sec)
__... Lo.._---'-_',
.1 111
. 11' L.o.._---'-_', _~.-----'--:'
Time (sec)
Fig. 3 . Control Torques
7. ACKNOWLEDGMENT The authors wish to acknowledge the support of the US National Science Foundation under Grant MSS-9057079 and Purdue University under a Purdue Research Foundation Grant .
8. REFERENCES Ambrosino, G., G. Celentano and F. Garofalo (1988) . Adaptive tracking control of industrial robots. Jour-
198