Proceedings of the 2001 IEEE International Conference on Robotics & Automation Seoul, Korea • May 21-26, 2001
Good Vibrations: A Vibration Damping Setpoint Controller for Continuum Robots Ian A. Gravagne∗, Christopher D. Rahn†, and Ian D. Walker∗
[email protected],
[email protected],
[email protected]
Abstract In this paper, we focus on a class of robotic manipulators that utilize continuous backbone structures. Such manipulators, known as “continuum” robots, exhibit behavior similar to tentacles, trunks, and snakes. Specifically, we have previously discussed some of the mechanical and kinematic details of the Clemson “Tentacle Manipulator.” This work examines the dynamic characteristics of this manipulator, proposing a vibration damping control strategy for configurations with the worst vibration characteristics. We begin by formulating the dynamics for one section of the Tentacle Manipulator. We then proceed to develop a new vibration control strategy which incorporates a setpoint regulator. We supplement the theoretical developments with experimental results.
1
pensive and complicated force feedback mechanisms. In principle, the preceding characteristics make continuum and hyper-redundant manipulators suitable for delicate or dangerous tasks where a traditional robot could not reach, or where single-point failures would paralyze subsequent operation of the robot. Examples of such tasks are toxic waste inspection and removal, and navigation or inspection of highly cluttered environments such as collapsed buildings.
Introduction
The study of hyper-redundant and continuum manipulators Þnds motivation from the natural world. Even from a qualitative point of view, manipulators such as tentacles and trunks exhibit special capabilities not shared with low degree-of-freedom (DOF) designs. While it is not yet possible to emulate real tentacles or trunks, artiÞcial continuum designs do possess several key desirable features, including improved obstacle avoidance capabilities and a signiÞcant transference of weight and complexity away from the actual manipulator, back to a more suitable location. We have previously termed this arrangement “remote actuation” [3], and it allows a substantial reduction in the overall design complexity (and cost) for continuum manipulators such as the Clemson Tentacle Manipulator (Þgure 1) and the Rice/Clemson Elephant’s Trunk Manipulator [3]-[6]. Continuum robots also possess another key feature, termed inherent compliance. In other words, the inÞnite-dimensional kinematics admit an inÞnity of possible backbone conÞgurations for any given Þnite set of applied forces or torques along the backbone. Thus, the robot will comply with environmental obstacles or non-conservative loads without the need for ex∗ Department of Electrical and Computer Engineering, Clemson University † Dept. of Mechanical and Nuclear Engineering, Pennsylvania State University.
0-7803-6475-9/01/$10.00© 2001 IEEE
Figure 1: The Clemson Tentacle Manipulator. The manipulator consists of 2 independent sections on a continuous backbone consisting of a thin elastic rod. In this paper we will focus on a simpliÞed dynamic description of the Tentacle Manipulator, which is a small 2section (four DOF) experimental prototype. Its backbone consists of a continuous elastic rod, with cable guides periodically spaced along its length. Four cable pairs run through the guide eyelets, two pairs terminating at the midpoint and two at the endpoint. Thus, torques may be applied in orthogonal directions at the midpoint or endpoint of the backbone by exerting tensions on the cables. However, because the backbone is relatively long and slender, sudden movements tend to excite undesirable vibratory motions. The objectives of this paper are to deduce a simpliÞed dynamical description of the backbone by applying a physically reasonable set of constraints and assumptions to the general 3-D dynamics for rods and beams; to apply a simple control law which
3877
will permit setpoint regulation of the midpoint and endpoint orientations while improving vibrational damping; and to illustrate the effectiveness of the controller by experimental implementation on the Tentacle Manipulator. In the context of traditional vibration control literature, the Tentacle Manipulator is a 3-dimensional variation on the “clamped-free” beam conÞguration. However, in contrast to the usual application of boundary shear forces in the clamped-free problem, here only boundary torques are available at the free end. The absence of controllable boundary shear forces distinguishes the current problem, along with our desire to regulate the section orientation to a non-zero setpoint and the multi-dimensional aspect of the dynamics. Several researchers have addressed topics related to manipulation and locomotion using hyper-redundant and continuum backbones. Among these, the pioneering work of Hirose [1] represented a large step forward in the realization of practical snake-like devices. Chirikjian and Mochiyama, [9]-[12], contributed to the theoretical development of hyper-redundant kinematics, path planning, and shape optimization. Further work in kinematics with a tilt toward practical design considerations appears in [2]-[6]. Another trunk-like prototype can be seen in [8]. Similar to the Elephant’s Trunk but signiÞcantly larger is the GreyPilgrim “Emma” serpentine manipulator [7]. Robinson and Davies [13] also provide a good overview of work in the area.
2
Rod Dynamics
In order to get the best kinematic properties in a remotely-actuated continuum device (such as maximized workspace and nearly constant curvature deformations), the friction between the cable and the cable guides must be minimized. This frictional component depends on conÞguration; in other words, it increases signiÞcantly as the tension in a cable increases and the backbone bends. Consequently, cable and stand-off friction is minimized near the zero-stress reference conÞguration, a straight line in this case. Unfortunately, while minimizing cable friction has beneÞts from a kinematic point of view, lack of friction prevents vibrations in the rod from damping out quickly after a commanded sequence of movements. The naturally antagonistic goals of good kinematic characteristics and fast vibration damping present a challenge to the motion control development, a later topic of this paper. Pursuant to the observation that vibrations present the biggest challenge near the straight reference conÞguration, the overall rod dynamics may be substantially simpliÞed. The following development justiÞes these simpliÞcations. We initially constrain the allowable conÞgurations of a rod of length L to a set of spatial curves C with
the following properties ª ¾ ½ © L] → 0 represents lumped-parameter energy stored in point masses and springs. ° ° ° ° Because, by the deÞnition of Q we know that °q i (s)° = 1, the Þrst condition combined with (1) implies that ¤T £ p0 (s) = x0 (s) y 0 (s) z 0 (s) = q 1 (s) (4) so that the curve tangent is aligned with (and in fact, equal to) the Þrst of the triad of orientation column vectors in Q. We will see the signiÞcance of this assumption shortly. For the second condition, we assume that any “dirac delta” distributions in f(s) are disallowed; intuitively this implies that the rod cannot experience single-point changes in distributed coordinates such as curvatures or positions. Condition 2 does not arise from strictly mathematical arguments; it simply states a physically reasonable assumption motivated by engineering judgement. We continue the dynamic development with derivations of the potential and kinetic rod energies.
2.1
Potential Energy
Consider an inÞnitesimal change in position on the beam from point s to s + ds. During this transition, the local rotational rate of change of the orientation Q is [Ω×] = QT Q0 [19]. Note [Ω×] is a skew-symmetric matrix whose associated axial vector is Ω; in other words Ω1 0 −Ω3 Ω2 0 −Ω1 ⇒ Ω = Ω2 (5) [Ω×] = Ω3 −Ω2 Ω1 0 Ω3
3878
Figure 2: An illustration of the frame Q(s) moving to Q(s + ds) with its associated local angular velocities. Each element of Ω represents the relative velocity with which each axis in Q is rotating as s moves to s + ds. Elements Ω2 and Ω3 represent two ßexural rotations, and Ω1 represents torsional rotation, or twist, as can be seen in Þgure (2). The potential energy in the rod takes the Figure 3: The beam in (A) experiences signiÞcant shear form [15] deformations because of its resistance to compression and Z L tension along the exterior walls. In (B) shear effects are © T ª 1 Ω K1 Ω + p0T QK2 QT p0 ds (6) not present because the concave side has shortened and PE = 2 0 the convex side as lengthened. K1 = diag{GJ, EI1 , EI2 } K2 = diag{EA, GA2 , GA2 } and W = QW QT 2
For simplicity (and practicality), the robots under consideration have circular cross-sectional backbones, so EI1 = EI2 = EI and GA1 = GA2 = GAs . In general, we refer to EI as the bending stiffness, GJ as the torsional stiffness, EA as the axial stiffness and GAs as the shear stiffness. The purely quadratic expression in (6) implicitly assumes small strains; given the aspect ratio (length to thickness) of the rods at hand, small strain is an accurate assumption [20]. (Note that small strains do not necessarily imply small deßections.) The second component of the potential energy describes energy due to shear deformations of the rod. However, the quantity p0T QK2 QT p0 = EA when (2) from condition 1 is taken into account. Thus shear deformations contribute nothing to the overall potential energy, and the axial stiffness contributes only a constant offset which may be neglected without loss of generality. In effect, condition 1 states that the inßuence of shear may be neglected. Figure (3) offers a pictorial representation of this condition. The robots under consideration in this paper have backbones corresponding to case (B) in Þgure (3). At this point we make an important connection with previous kinematic analyses. The potential energy may be written Z Z ª 1 L © 1 L T Ω K1 Ωds = tr [Ω×]T W1 [Ω×] ds 2 0 2 0 Z o 1 L n úT = (7) tr Q W2 Qú ds 2 0 ½ ¾ GJ GJ GJ , , where W1 = diag EI − 2 2 2
1
Expression (7) exactly duplicates the optimization measure used by Chirikjian and Gravagne in [3], [4] and [11], [12] for kinematic optimization and analysis, indicating the close connection between those works and the current model. Here, the weighting matrix W2 is conÞguration dependent and may not be positive deÞnite; it therefore would not have been an obvious choice apriori. However, this analysis illustrates that continuum kinematics tools and descriptions developed previously still apply to the current physically-motivated model.
2.2
Kinetic Energy
The complete derivation of the kinetic energy is too lengthy to cover here; however, the Þnal result makes intuitive sense. Essentially, we now assume that all quantities composing the potential energy now vary not only with arc length, but also with time. Having the same form as the local spatial rate of rotation seen earlier, the local angular velocity is [ω×] = QT Qú with Qú , dQ(s,t) dt . Associating the angular velocity vector ω with its skewsymmetric matrix [ω×] in the fashion described previously, we write the kinetic energy as Z ° °2 1 L Mρ °pú (s, t)° + ωT (s, t)Iρ (s)ω(s, t)ds (8) KE = 2 0
where Iρ (s) is the rotational inertia density tensor for the reference conÞguration and Mρ is the linear mass density of the rod. (Note that neither quantity varies with time.) Both of the expressions for kinetic and potential energy may be seen in similar form with slightly less restrictive conditions in [15].
3879
2.3
Work due Forces
to
In addition to potential and kinetic energy, which arise from conservation principles, energy may be injected or dissipated through the application of forces and torques. On the tentacle robot, there are two sets of cables in orthogonal planes which run down the backbone. Tension in a cable must balance with compressional forces in the beam itself, at every point along its length. Thus, the cable tension represents a distributed conservative input which can perform work. The tensions produce a distributed torque τ about the three axes in Q, proportional to the distance from the backbone to the cable. Since the cables are not of producing ¤ £ directly capable torsional torques, τ = 0 τ 1 τ 2 where τ 1 = F1 a(s), τ 2 = F2 a(s), F1 and F2 are the tensions in orthogonal planes, and a(s) is the distance from the cable to the backbone. So the work done to the backbone is Z L τ T Ωds. (9) W = 0
cβ = cα sβ sα sβ
Non-Conservative
−cα sβ −sα sβ s2α + c2α cβ −sα cα (1 − cβ ) −sα cα (1 − cβ ) c2α + s2α cβ
with sα , sin(α), cα , cos(α), etc., as in Þgure (4). Since the cables cannot produce twists, the angle α remains constant for any particular choice of cable tensions,¤ with £ [4] showing that τ = 0 F1 a cos(α) F2 a sin(α) . Next we note that the region of interest for vibration control corresponds to small β (i.e. β < 20◦ ). Given the restrictions on α and β, the no-shear condition of (2) combined with (10) yields dx ≡ x0 = cos β ' 1 ⇒ ds ' dx ds
(11)
For all practical purposes we allow ds = dx and, because x(0, t) = 0 ∀t, then x may replace s as the independent spatial variable. A glance back at Þgure (4) indicates the validity of this statement for “small” deßections. Consequently, we may Þnd the slope of the functions y(x) and z(x) approximated as angles
y 0 (x, t) = β(x, t) cos α and z 0 (x, t) = β(x, t) sin α. In previous work [4], we have discussed that the small (12) spatial variation of a(s) presents some practical design T 0 challenges but does not materially detract from the de- Evaluating [Ω×] = Q(s, t) Q(s, t) yields sired kinematic behavior. Thus we consider a as a design £ ¤T 0 β 0 cos α β 0 sin α . (13) Ω = constant hereafter. £ ¤T 00 00 0 y (x, t) z (x, t) = .
2.4
Simplifications Due to Small DeflecReferring back to (6) the new expression for the potential tions energy is
1 P E = EI 2
Z
L
(y 00 )2 + (z 00 )2 dx
(14)
0
Similarly, we make one further observation that the rotational kinetic energy contributes little compared with the translational kinetic energy, so referring back to (8) reveals Z 1 L Mρ (yú 2 + zú 2 )dx. (15) KE = 2 0 To summarize, these expressions for the system energy require the following simpliÞcations: negligible shear effects, relatively small deßections from the reference conÞguration, and that the backbone at any instant in time remains planar, or nearly planar.
Figure 4: An illustration of how the angles α(s) and β(s) contribute to the moving frame Q(s) at any instant in 2.5 Dynamical Equations time. Given the analysis this far, we employ the expressions for potential and kinetic energy and non-conservative work in This section concentrates on obtaining a tractable expresthe derivation of the dynamics via Hamilton’s Principle. sion for the dynamics of the beam near the straight conÞgWithout demonstration, the results are uration, which is the area of interest as discussed earlier. 00 First we must make an explicit choice for the orientation (16) [EIy 00 (x, t) − τ 1 ] + Mρ y¨(x, t) = 0 tensor Q(s, t). In [4] a we argued for the choice 000 0 EIy (L, t) = τ 1 (L, t) EIy 00 (L, t) = τ 1 (L, t) Q(s, t) = Q(α(s, t), β(s, t)) (10) 3880
with an identical expression for z(x, t). Similar expres- with sions exist in [14] under the subject of vibration damping using piezo-electric distributed actuators. Here, we have Kd argued that the time-varying distributed torque is spatially constant over [0, L] so the dynamics, with geometric boundary conditions appended, reduce the distributed Jef f torque to an equivalent boundary torque, EIy0000 (x, t) + Mρ y¨(x, t) EIy 000 (L, t) EIy 00 (L, t) y 0 (0, t) y(0, t)
= = = = =
0 0 τ 1 (L, t) 0 0
"
" # # 2 Br ab + akd akp = , Kp = , (23) kc + rb kc + rb " # 2 Jr ab = . kc + rb
(17a) (17b) (17c) (17d) (17e)
Because of the relatively high gear ratio r required to bend the rod, (20) suggests that the motor will feel little of the back-driving effect of the rod dynamics transmitted though the cable tension F1 . The coupling factor kc increases the effective back-driveability of the motor/gear system, providing greater control over the rod boundary. Associating (22) with (17c) results in the new boundary with an identical expression for z(x, t). The decoupled, linear mathematical form of (17), along condition with energy expressions (15) and (14), could probably EIy 00 (L, t) + Jef f y¨0 (L, t) = −Kd yú 0 (L, t) − Kp y˜0 (L, t) have been logically deduced without the need for the (24) extensive preceding derivations. However, in a multidimensional problem, there is always the potential for non-linear cross-terms or coupling between the general- 4 Controller Stability ized coordinates. Similar complexities here were nulliÞed through the careful application of certain assump- Proving the stability of the closed loop system requires tions and observations which may not have been obvious two steps: proving the boundedness of the total syswithout a derivation from Þrst principles. tem energy, and using an invariance principle to illustrate point-wise convergence of the rod to a time independent steady state. We begin by choosing a Lyapunov candidate
3
The Actuator Model
Without loss of generality, we may concentrate on the rod behavior in the x, y plane and assume identical behavior in the x, z plane since the dynamics are decoupled. Next, we note that there is a pulley of radius b to which the cables attach, driven by a motor through a gear ratio of r > 1. The angle of the motor is θm and the angle of the pulley is θ p . (The reader may refer ahead to Þgure 7.) Given that the motor has rotational inertia J and viscous friction B, we may take the simple motor model
Z
L
1 1 Mρ yú 2 +EIy002 dx+ Kp y˜0 (L, t)2 + Jef f yú 0 (L, t)2 2 2 0 (25) where V consists the distributed rod energy, plus discrete component energies from the effective motor inertia and a virtual spring attached to the free boundary. The power dissipated in this system can then be calculated as V =
J ¨θm + B θú m + τ p = τ m
1 2
Vú = −Kd yú 0 (L, t)2 ≤ 0
(26)
(18) after accounting for the dynamics in (17) and boundary condition (24). Thus boundedness of the system energy where τ p = rb F1 is the torque due to the cable tension, is proven. and τ m the torque generated by the motor itself. Note Next we wish to explore the possible system solutions that ú ≡ 0. This implies that yú 0 (L, t) ≡ 0, and to study this if V ra (19) case we Þrst note that the dynamics in (17) will admit θ m = rθ p = y 0 (L, t) b a separable solution, y(x, t) = Y (x)W (t). So yú 0 (L, t) = so that (18) becomes ú (t) ≡ 0 occurs in two cases. Y 0 (L)W ra 0 b ra 0 Case 1: Y 0 (L) = 0. This is simply the zero end-angle (20) J y¨ (L, t) + B yú (L, t) + F1 = τ m b b r situation. Admissible spatial solutions take the form Now we choose the feedback control law Y (x) = A cos(kx) + B sin(kx) + C cosh(kx) + D sinh(kx). τ m = −kp y˜0 (L, t) − kd yú 0 (L, t) − kc F1 (21) (27) 0 0 0 Manipulation using the boundary conditions indicates y˜ (L, t) = y (L, t) − yd A = − C and B = −D and with kp , kd , kc > 0 and yd0 the desired boundary angle setpoint. Substituting (21) back into (20) and solving for A(sin kL − sinh kL) + B(− cos kL − cosh kL) = 0 F1 produces A(− cos kL − cosh kL) + B(− sin kL − sinh kL) = 0 0 0 0 τ 1 = aF1 = −Jef f y¨ (L, t) − Kd yú (L, t) − Kp y˜ (L, t) (22) A(− sin kL − sinh kL) + B(cos kL − cosh kL) = 0 3881
Direct calculation reveals that det(M T M ) > 0 ∀k which implies that rank(M ) = 2 and the only solution to (28) is A = B = 0, and therefore C = D = 0. Thus this case indicates that y(x, t) = 0. ú (t) = 0. This implies that y¨(x, t) = 0, so Case 2: W the Þeld equation (17a) suggests Y 0000 (x) = 0. This means Y 000 (x) = c1 but Y 000 (L) = 0 implies c1 = 0. Therefore, Y 00 (x) = c2 which implies that Kp ye0 (L, t) = c2
20 kd = 0 kd = 0.04 15
Boundary Angle Error (Degrees)
The coefficients of A and B can be placed in a matrix M ∈
