Proceedings of the 2003 IEEE International Conference on Robotics & Automation Taipei, Taiwan, September 14-19, 2003
100
Mode Shape Compensator for Improving Robustness of Manipulator Mounted on Flexible Base Jun Ueda∗ and Tsuneo Yoshikawa∗∗ Nara Institute of Science and Technology (NAIST), Nara, 630-0192, Japan, E-mail:
[email protected] Department of Mechanical Engineering, Kyoto University, Kyoto, 606-8501, Japan, E-mail:
[email protected] ∗
∗∗
Abstract— In this paper, the concept of the ’robust arm configuration’ (RAC) is expanded using a mode shape compensator. This compensator improves the robustness of the arm configuration which is far out of the RAC. The compensator consists of a constant gain matrix and acceleration of each joint. The design method for this mode shaping matrix is presented based on the mode shaping algorithm. Effect upon the manipulability is also examined. It suggests that inclining of DME’s principal axes, resulting in changing the rigid body dynamics, by the compensator leads to the improvement of the robustness. The validity is confirmed by a numerical example performed with a 2 DOF planar manipulator mounted on a 1 DOF flexible base. A high bandwidth settling is realized with obtained compensator.
bility to great extent. In this paper, the concept of the ’robust arm configuration’ (RAC)[13] is expanded by using a mode shape compensator. It is a novel dynamic compensator which improves robustness using acceleration of joint axes. The compensator makes the system close to positive real as the RAC. A numerical design method using the measure which indicates the distance from the RAC is also presented. The validity of this approach is confirmed by a numerical example. Effects upon the change of manipulability is also examined.
Keywords— Flexible Robots, Flexible Base, Mechanical Resonance, Modal Analysis, Robust Control
II. Robust Arm Configuration of Manipulator A. Flexibility of the Base
I. Introduction The demand of high speed and accurate assembling is increasing with the advance of industrial products. One of the primary factors of limiting the bandwidth is the flexibility of the base on which a tool or a manipulator is mounted. Although many researches has been presented for manipulators with link or joint flexibility[1][2][3], this problem due to the base flexibility has hardly been studied. Simultaneous structure/control design has been studied[4][5][6][7][8] for this interaction problem. Recently, to overcome the ambiguity in mixing both design parameters, a passivity-based mechanical design [9][10][11] has been investigated. It is indicated that making the part of the frequency domain where passivity is satisfied as large as possible may improve the controllability and the robustness of the mechanical system itself. We have examined the controllability of rigid manipulator mounted on a flexible base by applying this mechanical design criterion to the evaluation of arm configuration. It was shown that there exists a set of arm configurations where high robustness and controllability are obtained. We defined this configuration as the ’robust arm configuration’ (RAC) where the linearized dynamics is positive real. A high-gain task-space control of the end effector can be applied in the neighborhood of the RAC based on the Lyapnov indirect method. However, there still remains an instability problem of the configuration which is far out of the RAC. A considerable solution is adding a viscosity with velocity feedback to the joint axes[12]. This method is valid for space-craft control. However, considering the application for robot manipulators, it deteriorates the manipula-
0-7803-7736-2/03/$17.00 ©2003 IEEE
A typical situation we are going to study is given in Fig. 1. A flexible assembly cell which consists of 2 degrees of freedom (DOF) planar rigid manipulator and two pallets on moving tracks is considered. As shown in Fig. 2, the parts are taken out from the parts feeder and are attached to the workpiece. Fig. 3 shows its dynamics model. The base driven by a ball screw on which the manipulator is mounted is considered as a passive visco-elastic joint. Since the control bandwidth of the ball screw is not high compared with the manipulator, the ball screw is positioned beforehand and is not driven during the assembly. During a motion, the root of the base twists due to the flexibility of the ball screw and the guide rods that support the base. In general, it is very difficult to attach additional sensors to measure the base flexibility. Compensation is through a task-space feedback control. The positioning error between the pallet and the end effector is measured directly using a narrow visual sensor attached on the pallet or the end effector. If the servo gain of the task-space control is increased, the full-closed loop is easily destabilized. B. Dynamic Equation The system consists of an m DOF flexible base and an n DOF non redundant manipulator is considered, which includes the system of Fig. 3 as a special case. τ denotes joint torque, q the joint axis displacement vector. The dynamic equation is as follows. τ = M (q)¨ q + h(q, q) ˙
(1)
τ = [τ Tp , τ Ta ]T ∈ N (N = m + n) where τ p = [ τ1 . . . τm ]T is torque generated by passive visco-
3011
101
elastic joints, τ a = [ τm+1 . . . τm+n ]T is actuator torque generated by motors. Similary, q = [q Tp , q Ta ]T ∈ N where q p = [ q1 . . . qm ]T and q a = [ qm+1 . . . qm+n ]T . The subscript p denotes the passive visco-elastic joints and a denotes the active joints. Note that q p cannot be measured directly. M (q) ∈ N ×N denotes the inertia matrix and h(q, q) ˙ are the centrifugal and the Coriolis factor. The effect of the gravity is neglected as it is considered unimportant. A task-space control by measuring the positioning error is necessary to compensate the uncertainty of the inverse kinematics of a manipulator and uncertain position of the moving tracks. In this paper, we focus on a task-space feedback control of the end effector using Jacobian transpose. Let r = r(q) ∈ n , the position of the end effector , rd as the ¯ p = [θ¯1 · · · θ¯m ]T as the equilibdesired position of r, and q q Tp , q Tad ]T where rium points of q p . r d is realized by q d = [¯ T ¯ q ad = [θ(m+1)d · · · θ(m+n)d ] and it satisfies r d = r(q d ). Let rv = rv (q a ) be the position of the end effector when the
Driven by Ball Screw Base Bridge
2DOF Planar Work
Manipulator
Fig. 1. Flexible Assembly System
Line1: Parts Feeder
Ball Screw
τi = kpi (θ¯i − θi ) − kvi θ˙i (i = 1, · · · , m)
(2)
where kpi and kvi are the stiffness and the viscous coefficient respectively. Hereafter, we examine the local asymptotic stability of ¯ p , q ad for the nonlinear system (1) around r d , (around q joint space) by its linearized model. In this regard, the centrifugal and Coriolis factor can be ignored since these terms are not related to the stability. Substituting (2) for (1), we obtain
0 τa
= M q¨ + D q˙ + K(q − q d )
(3)
= where D = diag (kv1 , · · · , kvm , 0, · · · , 0), K diag (kp1 , · · · , kpm , 0, · · · , 0) are the stiffness and the viscous matrix respectively. T Replacing x = q˙ T , (q − q d )T and y = r(q) − r(q d ) , n-inputs n-outputs state space representation in the work coordinate system is obtained: P (q d , s) = C(sI − A)−1 B (4) −1 −1 −M D −M K where A = I O O −1
M J Tv , C = I O J . J = , B = O ∂r/∂q T ∈ n×(m+n) , J v = ∂rv /∂qTa ∈ n×n are Jacobian matrices. We have assumed that this flexibility is not measured, then τ a is calculated as τ a = J Tv f , where f denotes the control input force in the taskspace calculated by a feedback controller K c (s): f = −K c (s)y. Note that we use P (q d , s) when we emphasize that P (s) is a function of q d , otherwise we use P (s).
Parts Feeder
Guide Rod
displacement of the base is zero, so that r(q d ) = rv (q ad ). It is assumed that the torque τ1 , · · · , τm generated by passive joints are working to settle to the equilibrium points θ¯1 , · · · , θ¯m :
Line2: Work
C. Modal Analysis AC Servo Motor
Fig. 2. Flexibility of the Base
I m3 3
l3
lg 3
y
l1
SB
lg1 m1 I1
t1
t2
q1 x
m2
lg 2
q3 t3
I2
q2 l2
Fig. 3. 2 DOF Manipulator with 1 DOF Base
By modal analysis, P (s) can be expressed as a linear sum of the rigid mode and the m vibration modes . A has in total 2(n + m) poles, 2n of these poles are zero, corresponding to the rigid mode, 2m conjugate complex poles correspond to the vibration modes. ¯ i (i = 1 · · · , m) be 2m + 1 distinct Let λ0 = 0,λi ,λ ¯ i correspond to the rigid eigenvalues of A. λ0 , λi and λ mode and the ith vibration modes respectively. We define matrix U and V using u which satisfy: Au2j−1 = λ0 u2j−1 = o, (j = 1, · · · , n), Au2j = u2j−1 , Au2(n+i)−1 = ¯ i u2(n+i) where λi u2(n+i)−1 , (i = 1, · · · , m), Au2(n+i) = λ
∗ U = u1 · · · u2n u2n+1 · · · u2(m+n) and V = col v ∗1 , v ∗2 , · · · , v ∗2(m+n) = U −1 . v ∗ denotes the complex conjugate transpose of v. Applying U , V to A, the modal
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102
analyzed transfer function[14] of P (s) is obtained as: P (s) R0
= =
m 1 1 R0 + Ri 2 2 + 2ζ ω s s ˆis + ω ˆ i2 i i=1 n
ˆ Cu2j−1 v ∗2j B = J v M
−1
J Tv
III. Mode Shape Compensator A. Improving Robustness Using Acceleration of Joint Axes (5) (6)
j=1
Ri
¯ i (Cu2(n+i)−1 )(v ∗ = −2Re λ B) (7) 2(n+i)−1
R0 corresponds to the rigid body mode and R0 is positive ˆ = [Mkl ] ∈ n×n (m + 1 ≤ k, l ≤ m + semi-deninite. M n) is a partial matrix of M . Ri is called residue matrix. Note that R0 is positive-semi-definete and rank(Ri ) = 1 at most from (7). In addition, ω ˆ i = |λi | , ζi = −Re(λi )/ |λi | is ˆ i represent the damping coefficient obtained where ζi and ω and the natural frequency respectively. D. Definition of Robust Arm Configuration
Let G(s) = sP (s) be the transfer function from the control input f to the velocity y: ˙ We have shown that G(s) is positive real if and only if Ri = RTi > 0[13]. Definition q d which satisfies λmin (G(q d , jω) + GT (q d , −jω)) = 0
(8)
is called the ’robust arm configuration’ (RAC) where λmin (G) denotes the minimum eigenvalue of G. The RAC is a set of special configurations where the linearized system is positive real. Note that the dimension of the RAC is not equal to that of the taskspace. Lyapnov indirect method and Passivity theory suggest that a local asymptotic stability for the original nonlinear system can be guaranteed with a finite but high feedback gain within a small area including the RAC. Measuring the flexible part and the inverse dynamics solution are not necessary. E. Robustness Measure of Arm Configuration Recall that if the rank(Ri + RTi ) ≤ 2, there exist only minimum and maximum eigenvalue. The system is positive real if rank(Ri +RTi ) = 1 and λmin (Ri +RTi ) ≥ 0. We have proposed the following robustness measure which indicates the distance from the RAC: wi (q) = λmin (Ri + RTi )
In general, without the presence of additional sensors; strain gauges, accelerometers and so on, it is assumed that the displacement of the flexibility, q p = [ q1 . . . qm ]T cannot be measured. The displacement of the active joints q a = [ qm+1 . . . qm+n ]T can be measured if encoders are attached. We assume that q¨a = [ q¨m+1 . . . q¨m+n ]T can be ideally measured. Set the joint input torque as: ˆ q¨a τ a = τ˜a + E
(10)
where τ˜ a is calculated by an error feedback scheme K c (s) shown in Fig. 4; τ˜ a = −J Tv K c (s)y Substituting (10) for (1), the following equation is obtained: 0 (11) = (M − E)¨ q + D q˙ + K(q − q d ) τ˜a
O m×m O m×n where E = . Equation (11) shows that ˆ n×n O n×m E the inertia matrix M is modified to (M − E). The second term of (10) modifies the vibration mode so that the the same property of the RAC, i.e., positiverealness, is obtained in a direct way. We named this type of compensator the ’mode shape compensator’. The calculation of the mode shape compensator is simple, since it consists of a constant gain matrix and acceleration of joint axes. ˆ is a constant gain matrix which is designed for each E ˆ the ’mode shaping matrix’ arm configuration. We call E corresponding to the mode shape compensator. It is clear that not all elements of M but only those who correspond to active joints can be modified. However, an appropriately ˆ improves robustness drastically as shown in the designed E next part. B. Design of Mode Shaping Matrix It is necessary that (M −E) = (M −E)T > 0 is satisfied to avoid instability since M = M T > 0 in original systems.
(9)
The characteristic of this measure is that wi ≤ 0 commonly holds and wi = 0 holds only on the RAC for ith mode. The configuration where wi is near 0 is preferable for better controllability. The advantages of this measure are as follows. First, it is easily calculated by n × n numerical computation of eigenvalues. Second, it clarifies the optimality of the arm configuration based on the positive realness. Third, it does not depend on the underlying control law. Forth, it identifies the robustness for individual mode. This measure is closely related to the robust stability margin based on the normalized coprime factorization [15]. It supports the optimality of the robust arm configuration in the case of considering a whole class of stable controller.
3013
rd
Manipulator t = M (q )q&& + h(q , q& )
tp
t i = kp i (q i - q i ) - kv iq&i i = 1,L , m
ta
q
r(q )
y
q&&a
flexibility
Eˆ t~a
Jv
T
f
-K c (s)
Controller
Fig. 4. Mode Shape Compensator and Controller
103
ˆ Since wi (q d ) = 0 holds on the RAC of the ith mode, E should be designed aiming at wi (q d ) → −0 . A mode shaping method[16] is applied for designing the mode shaping matrix where the natural frequency and the residue are originally used as representative parameters. For simplicity, suppose that m = 1. The property of ˆ = ˆ 1 ]T [ w1 ω the first mode can be represented as φ1 (E) where w1 is the distance from the RAC and ω ˆ 1 is the natural frequency. We set the desired natural frequency as ω ˆ 1d . ˆ is designed aiming at E ˆ → φ1d (E ˆd) = [ 0 ω ˆ 1d ]T φ1 (E)
(12)
ˆ=E ˆ so that n(n + 1)/2 elements (upper Recall that E triangle part) are independent. A design vector is defined T
n(n+1)
as ρ = [ˆ e11 , · · · , eˆ1n , eˆ22 , · · · , eˆ2n , · · · , eˆnn ]T ∈ 2 ×1 ˆ = [ˆ where E ekl ](1 ≤ k, l ≤ n) . Let φk1 and ρk be the representative parameters and the design vector after the kth iteration. The next design vector ρk+1 is obtained by updating ∆ρk as ρk+1 = ρk + ∆ρk
TABLE I Link Parameters
m1 m2 m3 I1 I2 I3
l1 l2 l3 lg1 lg2 lg3
0.2(m) 0.3(m) 0.25(m) 0.02(m) 0.15(m) 0.125(m)
the center of mass respectively. Assume that no singular configuration is passed while working. The working range is −π < θ2 < π ,0 < θ3 < π. The stiffness and viscous constant are kp1 = 3.0 × 104(Nm/rad),kv1 = 10(Nms/rad). The compensator, for the end effector (0.33,-0.25) shown in Fig. 5, is designed. Fig. 6 shows the measure w1 from the RAC without compensating action where the distance of the contour line is 0.015. We have shown that the robustness of this configuration is not high so that the closed loop system is destabilized [13].
(13)
+
∆ρk = g1 H k1 (φ1d − φk1 )
(14)
where g1 (> 0) is a scalar gain and = . It is difficult to obtain H k1 analytically, hence an estimation process is applied as follows. Let ∆ρ1 , ∆ρ2 , . . . , ∆ρl be small variations from ρk and the corresponding representative parameters ∆φ1 , ∆φ2 , . . . , ∆φl from φ1 where H k1
20.0(kg) 7.0(kg) 5.0(kg) 0.066(kgm2 ) 0.0525(kgm 2) 0.026(kgm2 )
∂φk1 /∂ρT
n(n + 1) ) ∆φj = H k1 ∆ρj (j = 1, 2, · · · , l ≤ 2
0.25
Work Fig. 5. Example Configuration
(15)
The estimated H k1 which minimizes the squared error of |∆φj − H k1 ∆ρj | is given by: + (Y − H k−1 X)X + H k1 = H k−1 1 1
0.33
(16)
n(n+1)
where X = [∆ρ1 ∆ρ2 . . . ∆ρl ] ∈ 2 ×l , Y = [∆φ1 ∆φ2 . . . ∆φl ] ∈ 2×l . It is not difficult to design the mode shaping matrix of the system which contains multiple (m ≥ 2) modes . A task priority based mode shaping method [16] can be applied. ˆ = wi is only appropriate One might argue that φi (E) for the purpose of improving robustness; a desired natural frequency should not be clearly given. However, in practical calculation, the natural frequency sometimes converges into an unrealistic frequency since the redundancy is too high. For this reason, a certain desired natural frequency is given. IV. Numerical Design Example A. 2 DOF Planar Manipulator Mounted on Flexible Base The system shown in Fig. 3 is considered. Table I shows the link parameters. θi is the joint angle of axis i, mi is the mass , li and Ii are the length and the moment of inertia of link i and lgi is the distance between the axis i and
B. Design of Mode Shaping Matrix eˆ11 eˆ12 ˆ = Recall E where eˆ12 = eˆ21 . Set ρ = eˆ21 eˆ22 ˆ d ) = [ 0 346.8 ]T and g1 = 0.02. [ˆ e11 eˆ12 eˆ22 ]T , φ1d (E The modify of the natural frequency is done by the compensation of the inertia. Therefore, the desired natural frequency is set fractionally higher than the original value to avoid a large mode shaping matrix. After 800 recursive calˆ= culation loops, the mode shaping matrix is obtained as E 0.12 0.079 0.58 0.013 ˆ = , for original M . 0.079 −0.15 0.013 0.104 As a result, (1,1) element is canceled approximately 20 %, practically, it is considered in the range of safty. Fig. ˆ 1 and λmin (M − E) re7 shows the transition of w1 , ω ˆ 1 are converged to the spectively. It shows that w1 and ω desired values, and simultaneously the robustness is improved. Additionally, (M − E) stays positive definite so that the system with the compensator maintains its stability. The residue matrix is modified and becomes close to positive semi-definite: −0.032 −0.075 0.036 0.048 → (17) R1 = 0.037 0.085 0.05 0.066 Fig. 8 shows the proposed measure w1 with the obtained compensator. In certain region, (M − E) is not positive
3014
104 0.05
0.07 0.068
0 0.066
0.064
-0.05
0.062
-0.1 0
200
400 Iteration Number
600
800
(a) Proposed Measure w1
0.06 0
200
400 Iteration Number
(b) λmin (
400
−
600
)
800
350
(a) 3-D Surface
300
w = -0.09
0.4
y (m)
w = -0.03
250 0
0
200
400 Iteration Number
600
800
(c) Natural Frequency Fig. 7. Design of Mode Shaping Matrix
-0.4 -0.4
0
0.4
is zero, a linearized input-output relation is obtained as:
0.8
x (m)
ˆ − E) ˆ −1 J Tv f y ¨ = J v (M
(b) Contour Plot Fig. 6. Proposed Measure of Primary Dynamics
Then the DME within f ≤ 1 satisfies: ˆ ˆ T −1 −T ˆ ˆ −1 ¨ ≤ 1 y¨T J −T v (M − E) J v J v (M − E)J v y
definite ,so that the system is not stable. However, in the neighborhood of the designed configuration, the robustness is successfully improved. A simulation is performed; a point to point control from (0.33,0.25) to (0.33,-0.25). While the arm is moving, jointlevel tracking control is applied to follow the generated path [0 – 0.1](s). Then the controller is switched to the taskspace feedback controller [0.1 – 0.4](s) in the neighborhood of (0.33,-0.25). The applied settling controller is a taskspace PD feedback: K c (s) = K p + K v s, then the actuator torque is calculated as: ˙ τ a = −J Tv (q a )(K p y + K v y)
(19)
(18)
where K p = diag(1.0 × 105 , 1.0 × 105 )(N/m), K v = diag(1.5 × 103 , 1.5 × 103 )(Ns/m). Fig. 9 shows the positioning error for each control scheme. Without the compensator, the system is unstable and oscillates. However, with the compensator, it satisfactorily settles. C. Effects upon Manipulability The mode shape compensator modifies the dynamics of the rigid body so that the vibration mode is shaped. As a result of this mode shaping, the robustness is improved. It is necessary to examine how much the manipulability of the rigid body changes. This effect is estimated by evaluating the dynamic manipulability ellipsoid (DME) [17] of 2 DOF rigid body part without its flexible base. Consider the relation from a control input f to a tip ¨ . Assuming that the displacement of the base acceleration y
(20)
Fig. 10 shows these ellipsoids for the whole range of motion. The distance of the contour line is 0.15. The dashed line denotes the original DME and the solid line denotes the modified one. Note that in Fig. 10, the desired natural ˆ 1 + 100 rad/sec and frequencies are determined as ω ˆ 1d = ω executed the calculation for 800 times. Robustness is improved and the positioning error stably converges for each configuration. In Fig. 10, the contour plot of the measure w1 without the compensator is superimposed. Fig. 10 reveals that the size of the ellipsoid itself does not drastically change because the mode shaping matrix is relatively small to the inertia matrix. However, the principal axes of modified ellipsoids tend to incline compared to the original ones. Furthermore, the inclination of the principal axes is large where the robustness is not high, while it is not large in the neighborhood of the RAC. These results indicate feasible design criteria of the compensator. That is, inclining the principal axes by a compensator leads to a good design. V. Conclusion In this paper, the concept of the ’robust arm configuration’ (RAC)[13] has been expanded. A mode shape compensator has been proposed which improves the robustness of the arm configuration which is far out of the RAC. The compensator makes the system close to positive real so that a high-gain feedback can be obtained just as in the neighborhood of the RAC. The compensator consists of a constant gain matrix and an acceleration of each joint. The
3015
105
-3
Positioning Error (m)
x y
Settling
Move
x 10 3
0
-3
0
0.1
0.2 Time (sec)
0.3
0.4
(a) Without Mode Shape Compensator -3
Positioning Error (m)
0.4
y (m)
w = -0.03
w = -0.09
0
x y
Settling
Move
x 10 3
(a) 3-D Surface
0
-3
0
0.1
0.2 Time (sec)
0.3
0.4
(b) With Mode Shape Compensator -0.4
Fig. 9. Settling Performance -0.4
0
0.4
0.8
0.6
x (m)
(b) Contour Plot
w = -0.05
w = -0.1
0.4
Fig. 8. Proposed Measure with Obtained Mode Shape Compensator
y(m)
0.2
structure is simple and easy to use. The method of designing the mode shaping matrix has been proposed based on the task priority mode shaping method. The validity of this approach was confirmed by a numerical example. Manipulability analysis suggests that inclining of DME’s principal axes by the compensator leads to the improvement of the robustness.
-0.2 -0.4
-0.4
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0
-0.2
0
0.2 x(m)
0.4
0.6
0.8
Fig. 10. Change of DME of Rigid Body (Whole Region)
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