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Variable Structure Control Design for Manipulators with Flexible-joints
Copyright © IFAC 12th Triennial World Congress, Sydocy, Auslralia, 1993
V ARIABLE STRUCTURE CONTROL DESIGN FOR MANIPULATORS WITH FLEXIBLE-JOINTS Danwei Wang, Ho Yeong Khing and Soh Yeng Chai School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263, Repuhlic o!Singapore
ABSTRACT In this paper, wc present a variable structure control design for robot manipulators with joint flexibility. Singular perturbation technique is used to study the two time scale property of the robot dynamics. A variable structure controller, incorporating a PD feedback is used to control the rigid-body dynamics. It is shown that the fast mode can be regarded as a distlll'bance to the rigid-body dynamics and hence a fast mode damping is optional in path tracking. Requiring only limited knowledge on bounds of robot dynamics, the controller is simple to design and implement. Keywords: Manipulators with flexible joints, Variable structure control.
tial energy function and g( q) = ;#q P( q) is t.he term describing the gravity effect.. The stiffness of the clastic joints are characterized by the diagonal positive definite matrix 1\. In the following, 11·111 denotes the I-norm, i.e., IIvlll = L7=1 Iv'!' In the case of a matrix, 11·111 denotes the induced I-norm, i.e., IIRII I = mm:j Li ITjjl.
I. INTRODUCTION
Path tracking control of a robot manipulator with joint flexibility has received considerable attention in recent years, see for instance [1-3]. Various control schemes have been proposed. However, these schemes invariably assumed the robot dynamics to be known exactly. In this paper, we present a control scheme which requires only rough bounds on the robot dynamics and disturbances. The usual model of a robot with flexible joint is first analysed using the standard singular perturbation technique to reveal the two time scale property of the dynamics. A simple derivative controller of the proposed composite controller is employed to damp out the elastic oscillations at the joints. Another components of the composite controller is used to control the rigid-body dynamics of the robot. This component controller has a linear feedback PD term and a nonlinear variable structure term. The nonlinear term is effective in overcoming the nonlinearity, coupling and distlll'bances which arc characterized by a bound in polynomial form. Due to the use of this nonlinear control, the PD feedback gains are not required to be very large. The nonlinear part of the controller are chosen for simplicity of design and implementation.
Suppose that wc have only limited knowledge of the robot dynamic parameters. The given desired trajectory is bounded by Ilqdlll ::; al and IIqdlll ::; a2' From [5], the inertia matrix is bounded by
for all q E Rn. Also it has been shown in [5J that the centripetal and coriolis term can be written as
where q appears in the n x n matrices Vi ( q), i = 1, ... , n, only as the argument of sine and cosine functions. Hence these matrices can easily be bounded, for all i = 1, ... , n,
11. ROBOT DYNAMICS AND MODELLING UNCERTAINTIES Consider the following robot dynamic equation:
M(q)ij
+ C(q, q)q + g(q) + 1\(q jO-1\(q-B)=u
0) = 0
(1) (2)
Following the definition of the gravity term, the corresponding bound is IIg( q) I1 ::; a4. We also assume that we have estimate of the st.iffness of the joint flxibility. In particular, we can wri te it as
In these 2n different.ial equat.ions, q E Rn is the vector containing n link angles, q is the link angular velocity vector, E Rn is the vector containing rotor angles, iJ is the rotor angular velocit.y vector, ami u E Rn is the control input.s applied at the n joint motors. Furthermore, M(q) and j arc respectively t.he inertia matrices of the manipulator aPoci the joint motors, while C'(q,q)q = [ftM(q)]q - ;#qrH'M('l)q] combines the cent. rifugal , coriolis terms, P( (1) is ih" poten-
o
where the f is a small positive number and 1\1 is a diagonal matrix wi th nonzel'O elements in the order of those in j. 421
Ill. DYNAMICS ANALYSIS AND CONTROL DESIGN
the boundary-layer equation:
Our objective is to design a simple feedback controller to drive the flexible joint robot manipulator to follow any given path, specified by qd and qd. We propose the following controller: 11
The fast mode control
U
=
Uf
+ Us
(15) Then the flexible-joint robot system dynamics can be expressed as equation (15) and
(3)
[M(q)
f is
+v
(5)
where K 3 = (kpI x
=
kvI)
Us
+ 1](t/E) + O(E).
(6)
(e(t») e(t)
with e = qd - q, k p , k v > 0, and v is a simple nonlinear function chosen as a polynomial in Ilx III
v = sgn(s)[,o
+'lllxlll +121Ixlli]
(18.b) (18.c)
(7)
With all these development, we can state the following result.
where sgn(s) = (sgn(s,), ... ,sgn(sn)f and the n-vector s is defined as, with a = kp/k v ,
s=(aI
THEOREM 1: Consider the closed-loop robot system (1) and (2) with the variable structure control (3)-(8). If the five controller parameters are chosen to satisfy k v > k p > 0, k v > m2/2, 10 > co, 1I > Cl and 12 > C2, then the path tracking errors converge exponentially to zero in the sense q(t) -> qd and q(t) -> qd.
(8)
I)x.
s vector then represents the switching hyperplane. Substitute the equation (3), (4) into (2), we have
Jjj
F
+ ~iJ + Ky = E
where y = 8 - q. Define z and we obtain
(16)
Note that the damping matrix for the fast mode can be set to zero and we still have a bounded 1](t/E) is bounded for bounded q(O) - 8(0). In this case, the 1] can be regarded as the disturbance to the rigid-body robot dynamics. With the estimate of the joint stiffness, we can specify the bound O(E) :s; a6. Use the norm bounds aj of the robot dynamics, we can define three positive constants as
where K 2 is diagonal matrix with nonzero elements in the order of those in J. The slow mode control Us is
= K 3x
=
The matrix K 2 is chosen so that (15) is asymptotically stable. This implies that 1](t/E) is bounded for bounded q(O) - 8(0), i.e., (17)
(4)
Us
+ J]ij + C(q,q)q + g(q)
= Ky
Jij
Us -
(9) Proof: Using control law (3)-(6) in dynamic equation (16),
and premultiply (9) by K
r z + 1-; z = 1-( J z.. + -;2. .. Us
we obtain the following error system i: = Ax
-
J") q
E
+ B(~A -
v)
(19)
(10) where
or (11) where K =
lft-.
M(q)ij Letting E state system
with
Also, equation (1) can be rewritten as
->
+ C(q,q)q + g(q) =
z
and
(12)
An = -[M(q)
0, equation (11) yields the quasi-steady-
+ J]-I(kvI + C(q,q) + a[M(q) + J]),
(13) Substitute this into (12), and
(14)
~A =
This is the limiting case where the joint stiffness is infinite, i.e., robots with rigid joints. From Tichonov's Theorem [6], z, q satisfy z(t) = 2(t) + 1](T) + O(E)
q(t) = ij(t)
[M(q)
+ J](ijd + ne) + C(q,q)(qd + ae)
+g(q) -1](t/E) - O(E).
(20)
With the given bounds on the dynamic parameters, we can write
+ O(E)
for t > 0, where T = tiE is the fast time scale, 1](T) satisfies
422
In the case :vhere _we have better estimates of M(q), C(q,q), g(q) as M(q), C(q,q), g(q) ,then the nonlinear control (5) can be replaced by
lIC(q, q)tj,dll = I1 L
q7'Vi (q)qdlll
u. = [M(q)
i=l
sL
+ J](qd + ae) + C(q,q)(qd + ae) + g(q) (22)
IlqJV;(q)qd - e7'V;(q)qd 11 1
With control law (22) in (16) and group the first three terms in ~A. It is easy to show that the bound coefficients Co, Cl and C2 can be substantially reduced. Therefore 10, 11 and 12 can be smaller to reduce the magnitude of the switching function. Further, since s = tK3x we have Ilsll S f11K31111xll. 50 IIxll --+ 0 implies Ilsll --+ O. Hence, the controll';.w defined in (3)-(8) is a sliding mode controller with the switching hyperplane defined by equation (8).
1=1
IIC(q,q)elll = 11 L(qJV;(q)e -
e7'Vj (q)e)111
i=l
Combining the above inequalities, we have a bound in polynomial form in llxlll as follows
IV. CONCLUSIONS A variable structure controller, together with a derivative component has been proposed for robust robot path tracking control of a flexible joint robot. The control design requires rough bounds on the uncertainties of complex robot dynamics and external disturbances. The controller is simple to design and implement.
where the Co, Cl and C2 are given in equation (18). Define a Lyapunov function as
where
REFERENCES
P _ ~ (2ak v I + a [M - 2 aiM + J] 2
+ J]
aiM + J]) [M + J] .
[11 5pong M W, "Modeling and Control of Elastic Joint Robots", A5ME Transaction, Vo1109, 1987,pp 310-319. [2] Khorasani K, 5pong W M, "Invariant Manifolds And Their Application To Robot Manipulators With Flexible Joints", Proc. IEEE Int Conf Robotics and Automation, 1985. [3] Ghorbel F, Hung J, 5pong MW, " Adaptive Control of Flexible Joint Manipulators", IEEE Control System Magazine, Dec, 1989, pp9-13.
This V(x) is a positive definite function in x because P is a symmetric and positive definite matrix as shown in [4]. Its time derivative along the robot motion is
v = x7'(A7'P + PA + P)x +2x7'PB(~A - v) = -x7'Qx
+ : (I\xf(~A -
v),
v
[4] Qu Z and Dorsey J "Robust Tracking Control of Robots by a Linear Feedback Law", IEEE Transactions on Automation and Control, Vo!. 36, NO. 9, 5ept 1991, pp 1081-1084. [5] Craig J J, Adaptive Control of Mechanical manipulator~, New York: Addison-Wesley, 1988. [6] Kokotovic P, Khalil H K, O'Reilly J,"Singular Perturbation Methods In Control: Analysis And Design", Academic Press, 1986.
where with
Q=
(a~I k~I)
>0
and the following equality, which is resulted from the properties of robot dynamics [5], has been used
for any vector y E Rn. Rewrite V as
V=
x7'(A7' P
+ PA + P)x + 2x7'PB(~A -
v)
+ s 7'(Ll.A - v) S _x7'Qx - [lIslll(,o + lll1 xlll + 1211 xlln - IIs1111l~Alld s _x7'Qx -lIslld('o - co)+ (,1 - cdllxlll + (,2 - c2)lI xlli] = _x7' Qx
S _x7'Qx. This implies that IIx 11 --+ 0 as t --+ 00. Obviously, the above implies that q(t) qd exponentially.
--+
qd and q(t)
--+
\l\l\l
423
V ARIABLE STRUCTURE CONTROL DESIGN FOR MANIPULATORS WITH FLEXIBLE-JOINTS Danwei Wang, Ho Yeong Khing and Soh Yeng Chai School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263, Repuhlic o!Singapore
ABSTRACT In this paper, wc present a variable structure control design for robot manipulators with joint flexibility. Singular perturbation technique is used to study the two time scale property of the robot dynamics. A variable structure controller, incorporating a PD feedback is used to control the rigid-body dynamics. It is shown that the fast mode can be regarded as a distlll'bance to the rigid-body dynamics and hence a fast mode damping is optional in path tracking. Requiring only limited knowledge on bounds of robot dynamics, the controller is simple to design and implement. Keywords: Manipulators with flexible joints, Variable structure control.
tial energy function and g( q) = ;#q P( q) is t.he term describing the gravity effect.. The stiffness of the clastic joints are characterized by the diagonal positive definite matrix 1\. In the following, 11·111 denotes the I-norm, i.e., IIvlll = L7=1 Iv'!' In the case of a matrix, 11·111 denotes the induced I-norm, i.e., IIRII I = mm:j Li ITjjl.
I. INTRODUCTION
Path tracking control of a robot manipulator with joint flexibility has received considerable attention in recent years, see for instance [1-3]. Various control schemes have been proposed. However, these schemes invariably assumed the robot dynamics to be known exactly. In this paper, we present a control scheme which requires only rough bounds on the robot dynamics and disturbances. The usual model of a robot with flexible joint is first analysed using the standard singular perturbation technique to reveal the two time scale property of the dynamics. A simple derivative controller of the proposed composite controller is employed to damp out the elastic oscillations at the joints. Another components of the composite controller is used to control the rigid-body dynamics of the robot. This component controller has a linear feedback PD term and a nonlinear variable structure term. The nonlinear term is effective in overcoming the nonlinearity, coupling and distlll'bances which arc characterized by a bound in polynomial form. Due to the use of this nonlinear control, the PD feedback gains are not required to be very large. The nonlinear part of the controller are chosen for simplicity of design and implementation.
Suppose that wc have only limited knowledge of the robot dynamic parameters. The given desired trajectory is bounded by Ilqdlll ::; al and IIqdlll ::; a2' From [5], the inertia matrix is bounded by
for all q E Rn. Also it has been shown in [5J that the centripetal and coriolis term can be written as
where q appears in the n x n matrices Vi ( q), i = 1, ... , n, only as the argument of sine and cosine functions. Hence these matrices can easily be bounded, for all i = 1, ... , n,
11. ROBOT DYNAMICS AND MODELLING UNCERTAINTIES Consider the following robot dynamic equation:
M(q)ij
+ C(q, q)q + g(q) + 1\(q jO-1\(q-B)=u
0) = 0
(1) (2)
Following the definition of the gravity term, the corresponding bound is IIg( q) I1 ::; a4. We also assume that we have estimate of the st.iffness of the joint flxibility. In particular, we can wri te it as
In these 2n different.ial equat.ions, q E Rn is the vector containing n link angles, q is the link angular velocity vector, E Rn is the vector containing rotor angles, iJ is the rotor angular velocit.y vector, ami u E Rn is the control input.s applied at the n joint motors. Furthermore, M(q) and j arc respectively t.he inertia matrices of the manipulator aPoci the joint motors, while C'(q,q)q = [ftM(q)]q - ;#qrH'M('l)q] combines the cent. rifugal , coriolis terms, P( (1) is ih" poten-
o
where the f is a small positive number and 1\1 is a diagonal matrix wi th nonzel'O elements in the order of those in j. 421
Ill. DYNAMICS ANALYSIS AND CONTROL DESIGN
the boundary-layer equation:
Our objective is to design a simple feedback controller to drive the flexible joint robot manipulator to follow any given path, specified by qd and qd. We propose the following controller: 11
The fast mode control
U
=
Uf
+ Us
(15) Then the flexible-joint robot system dynamics can be expressed as equation (15) and
(3)
[M(q)
f is
+v
(5)
where K 3 = (kpI x
=
kvI)
Us
+ 1](t/E) + O(E).
(6)
(e(t») e(t)
with e = qd - q, k p , k v > 0, and v is a simple nonlinear function chosen as a polynomial in Ilx III
v = sgn(s)[,o
+'lllxlll +121Ixlli]
(18.b) (18.c)
(7)
With all these development, we can state the following result.
where sgn(s) = (sgn(s,), ... ,sgn(sn)f and the n-vector s is defined as, with a = kp/k v ,
s=(aI
THEOREM 1: Consider the closed-loop robot system (1) and (2) with the variable structure control (3)-(8). If the five controller parameters are chosen to satisfy k v > k p > 0, k v > m2/2, 10 > co, 1I > Cl and 12 > C2, then the path tracking errors converge exponentially to zero in the sense q(t) -> qd and q(t) -> qd.
(8)
I)x.
s vector then represents the switching hyperplane. Substitute the equation (3), (4) into (2), we have
Jjj
F
+ ~iJ + Ky = E
where y = 8 - q. Define z and we obtain
(16)
Note that the damping matrix for the fast mode can be set to zero and we still have a bounded 1](t/E) is bounded for bounded q(O) - 8(0). In this case, the 1] can be regarded as the disturbance to the rigid-body robot dynamics. With the estimate of the joint stiffness, we can specify the bound O(E) :s; a6. Use the norm bounds aj of the robot dynamics, we can define three positive constants as
where K 2 is diagonal matrix with nonzero elements in the order of those in J. The slow mode control Us is
= K 3x
=
The matrix K 2 is chosen so that (15) is asymptotically stable. This implies that 1](t/E) is bounded for bounded q(O) - 8(0), i.e., (17)
(4)
Us
+ J]ij + C(q,q)q + g(q)
= Ky
Jij
Us -
(9) Proof: Using control law (3)-(6) in dynamic equation (16),
and premultiply (9) by K
r z + 1-; z = 1-( J z.. + -;2. .. Us
we obtain the following error system i: = Ax
-
J") q
E
+ B(~A -
v)
(19)
(10) where
or (11) where K =
lft-.
M(q)ij Letting E state system
with
Also, equation (1) can be rewritten as
->
+ C(q,q)q + g(q) =
z
and
(12)
An = -[M(q)
0, equation (11) yields the quasi-steady-
+ J]-I(kvI + C(q,q) + a[M(q) + J]),
(13) Substitute this into (12), and
(14)
~A =
This is the limiting case where the joint stiffness is infinite, i.e., robots with rigid joints. From Tichonov's Theorem [6], z, q satisfy z(t) = 2(t) + 1](T) + O(E)
q(t) = ij(t)
[M(q)
+ J](ijd + ne) + C(q,q)(qd + ae)
+g(q) -1](t/E) - O(E).
(20)
With the given bounds on the dynamic parameters, we can write
+ O(E)
for t > 0, where T = tiE is the fast time scale, 1](T) satisfies
422
In the case :vhere _we have better estimates of M(q), C(q,q), g(q) as M(q), C(q,q), g(q) ,then the nonlinear control (5) can be replaced by
lIC(q, q)tj,dll = I1 L
q7'Vi (q)qdlll
u. = [M(q)
i=l
sL
+ J](qd + ae) + C(q,q)(qd + ae) + g(q) (22)
IlqJV;(q)qd - e7'V;(q)qd 11 1
With control law (22) in (16) and group the first three terms in ~A. It is easy to show that the bound coefficients Co, Cl and C2 can be substantially reduced. Therefore 10, 11 and 12 can be smaller to reduce the magnitude of the switching function. Further, since s = tK3x we have Ilsll S f11K31111xll. 50 IIxll --+ 0 implies Ilsll --+ O. Hence, the controll';.w defined in (3)-(8) is a sliding mode controller with the switching hyperplane defined by equation (8).
1=1
IIC(q,q)elll = 11 L(qJV;(q)e -
e7'Vj (q)e)111
i=l
Combining the above inequalities, we have a bound in polynomial form in llxlll as follows
IV. CONCLUSIONS A variable structure controller, together with a derivative component has been proposed for robust robot path tracking control of a flexible joint robot. The control design requires rough bounds on the uncertainties of complex robot dynamics and external disturbances. The controller is simple to design and implement.
where the Co, Cl and C2 are given in equation (18). Define a Lyapunov function as
where
REFERENCES
P _ ~ (2ak v I + a [M - 2 aiM + J] 2
+ J]
aiM + J]) [M + J] .
[11 5pong M W, "Modeling and Control of Elastic Joint Robots", A5ME Transaction, Vo1109, 1987,pp 310-319. [2] Khorasani K, 5pong W M, "Invariant Manifolds And Their Application To Robot Manipulators With Flexible Joints", Proc. IEEE Int Conf Robotics and Automation, 1985. [3] Ghorbel F, Hung J, 5pong MW, " Adaptive Control of Flexible Joint Manipulators", IEEE Control System Magazine, Dec, 1989, pp9-13.
This V(x) is a positive definite function in x because P is a symmetric and positive definite matrix as shown in [4]. Its time derivative along the robot motion is
v = x7'(A7'P + PA + P)x +2x7'PB(~A - v) = -x7'Qx
+ : (I\xf(~A -
v),
v
[4] Qu Z and Dorsey J "Robust Tracking Control of Robots by a Linear Feedback Law", IEEE Transactions on Automation and Control, Vo!. 36, NO. 9, 5ept 1991, pp 1081-1084. [5] Craig J J, Adaptive Control of Mechanical manipulator~, New York: Addison-Wesley, 1988. [6] Kokotovic P, Khalil H K, O'Reilly J,"Singular Perturbation Methods In Control: Analysis And Design", Academic Press, 1986.
where with
Q=
(a~I k~I)
>0
and the following equality, which is resulted from the properties of robot dynamics [5], has been used
for any vector y E Rn. Rewrite V as
V=
x7'(A7' P
+ PA + P)x + 2x7'PB(~A -
v)
+ s 7'(Ll.A - v) S _x7'Qx - [lIslll(,o + lll1 xlll + 1211 xlln - IIs1111l~Alld s _x7'Qx -lIslld('o - co)+ (,1 - cdllxlll + (,2 - c2)lI xlli] = _x7' Qx
S _x7'Qx. This implies that IIx 11 --+ 0 as t --+ 00. Obviously, the above implies that q(t) qd exponentially.
--+
qd and q(t)
--+
\l\l\l
423