Experimental data shows vibration suppression with high efficiency. When no vibrations are present, our approach ensures effectively reactionless motion.
Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems October 9 - 15, 2006, Beijing, China
Singularity-Consistent Vibration Suppression Control With a Redundant Manipulator Mounted on a Flexible Base Toshimitsu Hishinuma and Dragomir N. Nenchev Department of Mechanical Systems Engineering Musashi Institute of Technology Tamazutsumi 1-28-1, Setagaya-ku, Tokyo, 158-8557 Japan
Abstract— This paper describes an experimental system for teleoperation of a redundant manipulator mounted on a flexible base. Kinematic redundancy is resolved with the help of an additional constraint, obtained from vibration dynamics. The problem of kinematic and algorithmic singularities is addressed via the Singularity-Consistent method developed in our previous research. Experimental data shows vibration suppression with high efficiency. When no vibrations are present, our approach ensures effectively reactionless motion. The stability of the system under teleoperation and while moving around algorithmic and kinematic singularities is also demonstrated.
I. I NTRODUCTION Many studies on macro-micro manipulators, or flexible-base manipulators, have focused on applications in the field of space robotics (see eg. [1]). In our previous research, we have developed a concept called the Reaction Null-Space which was applied to motion planning and control of a single-arm flexible base manipulator [2] and a dual-arm flexible base manipulator [3] to be used as space robots. Via the Reaction Null-Space, we decomposed the joint space such that dynamic decoupling between the macro and the micro subsystems is achieved. Thus, it was possible to suppress vibrations in the macro part in an efficient way, and also, to generate reactionless motions. Note, however, that in case of a single-arm flexible base manipulator, reactionless motion control implies that the endeffector motion is always confined to a specific path determined by the initial configuration. A few possible approaches to the problem, including the use of dynamic redundancy and/or kinematic redundancy, have been pointed out in [2]. The dynamic redundancy concept has been examined with the help of the dual-arm flexible base manipulator mentioned above. One of the arms was operated as a manipulation (loadcarrying) arm, while the other arm was used solely to compensate for the deflection of the flexible base. Unfortunately, with the dual-arm scenario one may encounter another problem: compensation maneuvers could lead to unavoidable collisions of the two arms. On the other hand, use of kinematic redundancy for simultaneous end-tip motion and base motion control has been proposed in [4], [5] with regard to a free-flying space robot. In [6], the redundant micro manipulator was used to compensate for the elastic deflections of the flexible-link macro part. In [7], the important problem of end-effector control in combination with vibration suppression control has been addressed. Vibration suppression control was derived from the null space of the manipulator Jacobian of the redundant micro part. In [2], we
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developed a control law for a redundant manipulator mounted on a flexible base, making use of the Reaction Null-Space. In this paper, we propose a simplified control law for a single-arm kinematically redundant flexible-base manipulator, aiming at simultaneous end-tip motion and vibration suppression control. It is shown that vibration suppression control is practically fully sufficient, yielding effectively reactionless motion, in the absence of vibrations. We address also the problem of encountering singularities. Note that the importance of this problem has been usually underestimated in literature, even in recent studies [8]. The problem is important, because besides kinematic singularities, there are also so-called algorithmic singularities. Such singularities are due to the imposed vibration suppression constraint, and are located inside the workspace. As noted in [9], it is physically impossible to realize vibration suppression at such manipulator configurations. Here, we tackle the problem by implementing the Singularity-Consistent method developed in [10], with application to a redundant arm, as in [11]. Thus, we obtain a unified framework for teleoperation of a flexible-base manipulator, able to handle both algorithmic and kinematic singularities. II. BACKGROUND A. The Reaction Null-Space Concept The equation of motion of a manipulator mounted on a flexible base can be written in the following form [2]:
H bm (xb , q) x ¨b Db x˙ b + q¨ 0 H m (xb , q) ˙ K b xb c (x , x˙ , q, q) 0 + + b b b = , ˙ 0 cm (xb , x˙ b , q, q) τ
H b (xb , q) H Tbm (xb , q)
(1)
where xb ∈ k denotes the positional and orientational deflection of the base with respect to the inertial frame, q ∈ n stands for the generalized coordinates of the arm H b , D b , and K b ∈ k×k denote base inertia, damping and stiffness, respectively. H m ∈ n×n is the inertia matrix of the arm. H bm ∈ k×n denotes the so-called inertia coupling matrix. cb and cm are velocity-dependent nonlinear terms, and τ ∈ n is the joint torque. No external forces are acting neither on the base nor on the manipulator. Under the simplifying assumptions, described in [2], the equation of motion can be linearized around the equilibrium
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of the base, as follows: ¨ b + Db x˙ b + K b xb = −H bm q ¨. H bx
(2)
Then, choose the control acceleration as ˙ b + (I − H + q¨ = H + bm Gb x bm H bm )ζ
(3)
n×k where Gb is a positive definite matrix, and H + bm ∈ denotes the Moore-Penrose generalized inverse of the inertia coupling matrix, I denotes the unit matrix of proper dimension, and ζ is an arbitrary vector. Then, since H bm H + bm = I and H bm (I − H + bm H bm ) = 0, it becomes apparent that controlled damping can be achieved by a proper choice of matrix Gb . Note that the second term on the RHS of the above equation stands for the Reaction Null-Space. In [2], the term was used to ensure the desired end-effector motion constraint. Below, we show that the desired end-effector motion can be realized without the Reaction Null-Space term.
B. Singularity-Consistent Method
(5)
where t is the normalized end-effector twist and q˙∗ is the twist magnitude. With the help of the column-augmented Jacobian ˜ = J −t , we rewrite Eq. (5): J q˙ ˜ J = 0. (6) q˙∗ This homogeneous equation has an infinite number of so˜ lutions that are obtained from the null-space of matrix J. When the manipulator Jacobian J is square (a nonredundant manipulator with m = n), there is just one nonzero vector in ˜ ) that can be expressed as: the null-space N (J T ˜ = nT det J , n (7) where n = (adjJ )t,
(8)
adjJ denoting the adjoint matrix of J. The singularityconsistent joint velocity is obtained then as q˙ = bn
A well known method for resolving kinematic redundancy is to impose an additional constraint [13]. We derive such an additional constraint, expressed in terms of joint velocity, from the vibration suppression condition. For this purpose, we integrate the control acceleration for vibration suppression in Eq. (3), without the Reaction Null-Space term: ¯ Gb xb . q˙ ≈ H bm
(9)
where b is an arbitrary scalar, regarded as control input. Choosing b = q˙∗ / det J, it will be possible to move the endeffector with the desired velocity ν = q˙∗ t. When approaching
(10)
Thereby, we assumed matrix H + bm , which is a function of the joint angles, to be constant (in the above equation denoted by the overbar). This is justified by the fact that the change of base displacement xb is much faster than that of the joint angles. The additional constraint is derived from the last equation: H bm q˙ = Gb xb .
(4)
The equation can be rewritten as: J q˙ − q˙∗ t = 0,
III. R EDUNDANCY R ESOLUTION T HROUGH V IBRATION S UPPRESSION C ONSTRAINT
+
We aim to control both end-tip motion and flexible base vibrations. It is inevitable that we will encounter thereby both kinematic and algorithmic singularities, the former due to rank deficiency in the manipulator Jacobian J(q) ∈ m×n , and the latter due to rank-deficiency of the inertia coupling matrix H bm . We will tackle these singularities in an unified way, with the help of the Singularity-Consistent method [10]. Denote by ν ∈ m the manipulator end-effector twist: ˙ ν = J q.
a kinematic singularity, b is assigned a proper constant value determined from motion continuity condition, so that at the singularity, the end-effector comes instantaneously to rest: q˙∗ = ¯b det J = 0. When applying the method to teleoperation (as we intend to do), the value of b around the singularity is modified as b = σ¯b, where σ denotes a sign variable, and ¯b > 0 is constant. The sign variable is needed to ensure continuous motion through the singularity, see [12] for details.
(11)
Let us assume now that the dimension k of the base deflection space equals the degree of redundancy of the manipulator, that is k = n − m. Combining the imposed end-effector velocity constraint Eq. (4) with the above additional constraint, we obtain: ν (12) = J vs q˙ Gb xb T where J vs = J T H Tbm ∈ n×n . The joint velocity can be then written as: ν −1 q˙ = J vs . (13) Gb xb
IV. T HE S INGULARITY-C ONSISTENT S OLUTION Though the above solution for the joint velocity vector was obtained in a straightforward manner, we must note that performance will inevitably degrade when matrix J vs becomes singular. From studies on kinematically redundant manipulators [13] it is well known that besides kinematic singularities where det J = 0, also so-called algorithmic singularities will appear which are due to the imposed additional constraint. Since the additional constraint used here is the vibration suppression constraint, we can expect that the capability to suppress vibrations will deteriorate around these algorithmic singularities [9].
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To cope with the singularity problem, we will rewrite the above joint velocity solution according to the SingularityConsistent method, with a slight modification: we keep the original end-effector twist representation. Then, following the procedure described in Section II, we compose the columnaugmented Jacobian as follows: J −ν ˜ J vs = (14) ∈ n×(n+1) H bm −Gb xb The homogeneous equation is written as: q˙ ˜ J vs = 0. 1
Fig. 1.
y High stiffness direction
˜ vs ) that There is just one nonzero vector in the null-space N (J can be expressed as ˜ vs n
= nTvs
det J vs
m6
l3 m5
T
T T
−(Gb xb ) where nvs = adjJ vs −ν consistent solution is then obtained as: T
TREP-R experimental setup.
(15)
(16)
m4
. The singularity-
q˙ = bvs nvs ,
m3
(17) m0
where bvs is an arbitrary scalar. Note that nvs is a function of both end-effector twist ν and base deflection term Gb xb . It is convenient then to rewrite Eq. (17) as:
m1
m2 q1
l g1
q2
l g2
l2
x Low stiffness direction
l1 Flexible base Fig. 2.
q˙ = bvs (nm ν + nb Gb xb ),
q 3 l g3
Model of TREP-R.
(18)
where nm ∈ n×m , nb ∈ n×k . According to the Singularity-Consistent method, away from a singularity, bvs can be chosen as bvs = 1/ det J vs which would guarantee both end-effector motion and vibration suppression control. On the other hand, around a singularity, choose bvs = ¯b, where ¯b denotes a constant value determined from motion continuity conditions. This means that (i) the end-effector will come instantaneously to rest at the singularity, and (ii), the vibration suppression capability will gradually degrade while approaching a singularity.
B. Dynamic model of TREP-R The system model is depicted in Fig. 2. The parameters of the arm and the base parameters are presented in Tables I and II, respectively. The natural frequency, damping ratio and base stiffness were estimated using the logarithmic decrement method applied to damped free vibration experimental data. The values obtained for the natural frequency and the damping ratio are ωn = 12.66 rad/s and ζ = 0.011, respectively. C. TREP-R control
V. I MPLEMENTATION OF THE M ETHOD A. Experimental setup Our experimental setup, called TREP-R, is depicted in Fig. 1. It is a planar system consisting of a 3R rigid arm manipulator attached to the free end of an elastic double beam representing the flexible base. Due to the specific design, we can assume that the base deflects only from the longitudinal axis, in the low-stiffness direction. Hence, the reaction moment and the reaction force component along the transverse axis of the base can be neglected as a disturbance. The deflection of the flexible base is measured by a strain gauge. A joystick is used to generate end-tip commands for the manipulator. The motor controllers work in velocity control mode.
There is one degree of redundancy since the manipulator has three-DOF (n = 3) and we control the position of the end-tip (m = 2). This means that k = 1. Indeed, as already explained, base deflection from the longitudinal direction only is considered. Matrix J vs is then 3 × 3. The additional constraint for vibration suppression, as in Eq. (11), becomes: hbm q˙ = gb xb .
(19)
Note that because we consider one-DOF base deflection, the gain gb and the base deflection xb are expressed as scalar values. The inertia coupling matrix hbm is represented as a 1× 3 row-matrix. The column-augmented Jacobian matrix from
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TABLE I
where K p = diag(kp , kp ) denotes a constant feedback gain matrix, and p stands for the end-tip position calculated from sensor data via the direct kinematics (DK). Note that all sensor data are filtered through a low pass filter. As already mentioned, away from a singularity, bvs is set to the inverse determinant, and the two sigma’s hold the sign of the determinant. In the vicinity of a singularity (noncritical), bvs is modified as bvs = ¯b, where ¯b > 0 is a constant determined from continuity reasons. When crossing a singularity, the sigma’s will be modified, as explained below.
M ANIPULATOR LINK PARAMETERS l1 , l2 , l3 lg1 , lg2 , lg3 m1 , m3 , m5 m2 , m4 m6
0.1 0.05 0.025 0.285 0.095
[m] [m] [kg] [kg] [kg]
TABLE II F LEXIBLE BASE PARAMETERS Length Height Width Thickness Tip mass hb (≡ m0 ) Stiffness kb Damping db
p
. pd
[m] [m] [m] [m] [kg] [N/m] [Ns/m]
. + pref σ m +
s
nm +
Four experiments have been performed, as follows: 1) Vibration suppression at a nonsingular configuration. 2) Vibration suppression at an algorithmic singularity. 3) Teleoperation with vibration suppression. 4) Teleoperation with enhanced vibration suppression. In all experiments, we used a constant bvs throughout the motion: ¯b = 1 kg−1 m−3 . The gain for vibration suppression gb was set to 20 kgs−1 , and the position feedback gain kp was set to 50 s−1 .
nb
+
q bvs σb
Fig. 3.
VI. E XPERIMENTS
DK
− kp + p d 1
0.4 0.1 0.09 1 × 10−3 0.45 191 0.33
TREP−R xb
A. Vibration suppression at a nonsingular configuration
gb
Singularity-consistent vibration suppression controller.
Eq. (12) is written as: ˜ vs = J J hbm
−ν ∈ 3×4 . −gb xb
(20)
˜ vs ∈ 4 in the kernel of the There is just one nonzero vector n above matrix. Then, the singularity-consistent joint velocity can be written as: q˙ = bvs nvs
(21)
where nvs ∈ 3 is composed of the first three components ˜ vs . Next, we separate the manipulator and base of vector n variables, as in Eq. (18): q˙ = bvs (σm nm ν + σb nb gb xb ),
(22)
where nm ∈ 3×2 and nb ∈ 3 . In addition, we introduced two separate sign variables (the sigma’s), which will be needed to control the behavior when moving through a singularity. Figure 3 shows the block diagram of the singularityconsistent vibration suppression controller for TREP-R. We obtain the desired end-tip velocity p˙ d via the joystick command. The velocity is integrated numerically to obtain the desired end-tip position pd . These values are used to calculate the reference end-tip velocity via proportional feedback: ν ≡ p˙ ref = K p (pd − p) + p˙ d
(23)
In the first experiment, the aim is just to demonstrate that the controller is able to compensate for end-tip position deviations during vibration suppression. In other words, vibration suppression will be ensured just by pure selfmotion. The initial manipulator configuration was set to [2.01, −1.85, 1.02]T rad — a stationary, nonsingular configuration. An impulsive external force was then applied to the flexible base. As can be seen from Fig. 4 (a), the base deflects just for a short time interval (solid line). The vibration has been suppressed almost instantly by the joint motion depicted in Fig. 4 (b). Note that in Fig. 4 (a), the free vibration graph has been superimposed for comparison (the dashed line). Figures 4 (c) and (d) display position coordinates of the endtip and the center of mass (CoM), respectively. It is apparent that the end-tip is just slightly disturbed, despite the relatively large disturbance force at the base. B. Vibration suppression at an algorithmic singularity The aim of the second experiment is to demonstrate deteriorated vibration suppression capability in the vicinity of an algorithmic singularity. The manipulator configuration was set to [2.53, −1.03, −0.87]T rad, as shown in Fig. 5. In this configuration, the CoM cannot accelerate in the −x direction, hence the singularity. The experimental data are shown in Fig. 6. From Fig. 6 (c) it can be confirmed that the determinant is almost zero. Two external impacts have been applied; the first one just before t = 2 s, and the other one at t = 6 s. From the base deflection graph (Fig. 6 (a)) it is seen that the vibration suppression capability has deteriorated significantly, when compared to Fig. 4 (a).
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Possible motion x
Fig. 5.
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Data from Experiment 1.
Fixed end−tip (self−motion)
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C. Teleoperation with vibration suppression With this experiment, we demonstrate the vibration suppression capability during end-tip teleoperation via a joystick. The initial manipulator configuration is a nonsingular one: [2.04, −1.01, 1.36]T rad. The graph in Fig. 7 (a) shows the position coordinates of the end-tip during joystick teleoperation. From Fig. 7 (b) it becomes apparent that the flexible base deviates significantly at about t = 3 s, which is due to a large impulsive force applied to the base. The manipulator links react immediately (Fig. 7 (d)) to compensate for any end-tip deviation (Fig. 7 (a)). In addition, this experiment demonstrates the singularityconsistency capability of the controller with regard to an algorithmic singularity. As seen from the determinant graph in Fig. 7 (e), an algorithmic singularity has been encountered at approx. t = 6 s. Also, it is apparent that the singularity has been crossed, since the sign of the determinant has changed. The smoothness of all graphs around that point suggests that this happened without any instabilities. Note however, that the end-tip was “reflected” backwards (see Figs. 7 (a) and (f)). This phenomenon has been discussed elsewhere [11]. Note also, that while σb has to change with the sign of the determinant, σm shouldn’t do so. Otherwise, the end-tip will start to oscillate around the algorithmic singularity [12].
The phenomenon of a reflecting end-tip around an algorithmic singularity, described above, is highly undesirable. Note that such singularities are located inside the workspace. This means that the effective workspace will be significantly reduced. To alleviate the problem, we propose below a modification of the control law. The algorithmic singularity encountered in the previous experiment occurred because it was not possible to accelerate the CoM in the direction of motion of the end-tip. In this experiment, we will relax the vibration suppression constraint and enable the CoM to do so. This, of course, means that the flexible base will necessarily deflect. Note, however, that the vibration suppression capability will not be disabled. Hence, any possible vibrations will be suppressed, as usual. We ensure such behavior with the following control law modification: q˙ = bvs [σm nm ν + σb nb (gb xb + we p˙ xd )],
(24)
is the x component of the desired end-tip velocity, where and we is a weighting factor. Experimental data are shown in Fig. 8. Hereby, exactly the same initial conditions were used as in the previous experiment. The weight we in the above control was set to 20 kg. In this experiment, no external force was applied. From the graph in Fig. 8 (c) it should be apparent that the CoM starts accelerating in the low-stiffness direction (x axis) at around t = 1.5 s. The base deflects accordingly (cf. Fig. 8 (b)). Any possible follow-up vibrations have been suppressed, though. For comparison, we superimposed the base deflection graph obtained with the same joystick input, but resolved through the manipulator Jacobian pseudoinverse. The follow-up vibrations are clearly seen. In addition, at the final stage, large amplitude vibrations were generated because the arm reached the vicinity of the kinematic singularity at the outer workspace boundary (see also Fig. 8 (f)). Comparing the end-tip motion traces in Figs. 7 (f) and 8 (f), it is seen that initially, they are similar. In the present experi-
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Fig. 8.
ment, however, the end-tip was not reflected. The algorithmic singularity was effectively shifted to the outer workspace boundary with the help of the enhanced vibration suppression control law (24) (cf. Fig. 8 (e)). VII. C ONCLUSIONS We have shown that kinematic redundancy resolution for a manipulator mounted on a flexible base, through a vibration suppression constraint, can play an important role. The effect achieved is the same as that of reactionless path motion control, developed in our previous studies. An advantage of the new method is that such a reactionless motion can be practically ensured within the entire manipulator workspace. This was possible because we addressed the important problem of handling algorithmic and kinematic singularities, by implementing the Singularity-Consistent method. With this method, both stable autonomous operations and stable teleoperation through the entire workspace can be guaranteed. Another advantage of the method is its simplicity. Also, the developed equations are general in the sense that they can be directly applied to spatial manipulators. Note, however, that the vibration suppression capability depends on the kinematic structure and on the mass/inertia distribution. Therefore, implementation details will have to be considered for each specific spatial structure. R EFERENCES
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