A Passivity-Based Control Scheme for Robotic Grasping and Manipulation Stefano Stramigioli1 , Claudio Melchiorri2 , and Stefano Andreotti2 1
2
Delft University of Technology Dept. of Information Tech. and Systems, P.O. Box 5031, NL-2600 GA Delft, NL E-mail:
[email protected] WWW: http://lcewww.et.tudelft.nl/~stramigi Abstract This article presents and discusses the application of an impedance control strategy to robotic grasping tasks. This control strategy, that can be used for both tips and full grasps, is based on two fundamental aspects. The first is the so-called ‘virtual object’ concept, by means of which the modalities of interaction between the fingers and the real objects can be properly defined. The second is the ‘damping injection’ principle, which allows a stable execution of the planned task. Major advantages of the presented strategy are the physical intuition on which it is based and its passive nature and which ensures stability in all the situations, including the transition from no contact to contact and vice versa. Experimental results will show the effectiveness of the control strategy.
1 Introduction It is well known, after the pioneering work of Hogan back in the 80’s [1], that the problem of controlling a robotic system interacting with the environment may be considered in the context of defining the desired dynamic relation between the interaction forces and velocities. Within this general framework, impedance control, allowing the implementation of such a dynamic relationship, can be considered as the reference control scheme. With respect to known manipulation and grasping control strategies, on the other hand, one of the main issues is how to deal with the control of the contact forces applied by the fingers to the grasped object. Although several approaches have been proposed in the literature to deal with this problem [2], very few among them consider explicitly the problem of controlling, or defining, the dynamics of interaction. In addition, one of the most problematic phenomena in force control strategies is that stability cannot be ensured if not assuming as known important features of the object to be grasped, like its stiffness and friction. Furthermore, a force control strategy alone is not suitable to properly control the transition between nocontact and contact. This is due to the obvious fact that force control is meaningful only in contact, since no force can be exerted in free space. For these reasons, a grasp control strategy based
DEIS, University of Bologna Via Risorgimento 2 40136 Bologna, Italy E-mail:
[email protected] WWW: http://www-lar.deis.unibo.it on physically-based observations and on passivity concepts seems worth to be pursued. Such an impedance strategy does not have the shortcomings of other grasping techniques: it is strictly passive in steady-state situations and the supplied energy in moving tasks is directly controllable. Moreover, the related compliance control of each finger allows for rolling, slipping, and whole-hand grasps in a natural way. The design philosophy behind these control synthesis methodologies is here indicated as Physical controller design [3] and follows Hogan’s Physical Equivalence Principle [1]. The paper is organized as follows. In Sec. 2 some background material on spatial impedance is presented, and in Sec. 3 the main features of the control scheme are presented. Sec. 4 reports and discusses some experimental results obtained with a laboratory setup, and Sec. 5 concludes with some remarks. 2 Physical Intrinsic Passive Control The proposed control scheme is physically interpretable in terms of mechanical 3D elements like springs, masses and dampers. Before presenting the complete control strategy, [3], a general notation for describing these 3D elements is introduced, see also the Appendix. 2.1 Springs: Spatial Compliance A spatial compliance is a geometric spring [4] connecting two rigid bodies Bi and Bj . Lonˇcari´c [5, 6] studied geometric springs represented by potential energy functions of the relative position of the rigid bodies to which they are attached. Fasse and Breedveld [7, 8, 9] and Stramigioli [4] extended this work. A spring between two rigid bodies Bi and Bj is characterized by means of a positive definite function representing the stored potential energy1 with the following form: Vi,j : SE(3) → ; Hij → Vi,j (Hij ) where Hij ∈ SE(3) is the matrix which represents the isometry which brings a chosen reference frame Ψj 1 This
function defines implicitly the unit of energy.
(fixed to body Bj ) to another reference frame Ψi (fixed to body Bi )2 . The wrench applied on the spring connecting Bi to Bj by body Bi and expressed in frame Ψj , with a relative position Hij is (see Notation): j f˜i,j nji,j j j j ∗ ˜ Wi,j (Hi ) = = RH j dVi,j (Hi ) i 0 0 j where f˜i,j ∈ 3×3 is an antisymmetric matrix correj sponding to the force fi,j ∈ 3 and such that: x1 0 −x3 x2 x = x2 ⇒ x˜ = x3 0 −x1 (1) x3 −x2 x1 0
˜j ∈ We can also associate to the wrench matrix W i,j 4×4 the corresponding vector representation which we j j T T := ((nji,j )T (fi,j ) ) ∈ 6 . It indicate with Wi,j j i = −Ad∗H j Wi,j is the can be seen thatWii = −Wi,j i wrench that the spring applies to body Bi expressed j i in the frame Ψi and that Wjj = −Wj,i = −Ad∗H i Wj,i is j nothing else than the wrench that the spring applies to body Bj expressed in frame Ψj and furthermore Wi,j = −Wj,i due to the nodicity of a spring [4]. A desired energy function can be defined (and implemented with the control) such that the relative configuration Hij = I4 corresponds to a minimum of the potential energy Vi,j (·) [7, 4]. In this configuration, the frames Ψj and Ψi will coincide. The energy function can be chosen such that the common origins of Ψi and Ψj in the equilibrium position represents the center of stiffness [4]. Expressed in the equilibrium frame (Ψi = Ψj ), we can then choose three 3 × 3 desired stiffness matrices Ko , Kt and Kc , corresponding respectively to the orientational, translational and coupling stiffnesses. From these stiffness matrices, we can calculate the so-called co-stiffness matrices [7, 4] Go , Gt and Gc related to the stiffnesses matrices by: 1 Gα = tr(Kα )I − Kα 2 where α = o, t, c and where tr() is the tensor trace operator. It is then possible to give an expression of the a function of the relative configuration wrench Wii as j j p R i i (see [7, 4]): Hij = 0 1 i f˜i nii j i Wi (Hi ) = where 0 0 nii = −2 as(Go Rij ) − as(Gt Rji p˜ji p˜ji Rij ) − 2 as(Gc p˜ji Rij ) f˜ii = −Rji as(Gt p˜ji )Rij − as(Gt Rji p˜ji Rij ) − 2 as(Gc Rij ) (2) 2 Note that H j is also the matrix expressing the change of i coordinates from Ψi to Ψj , but the corresponding motion is given by its inverse (Hij )−1 .
and where as() is an operator which takes the antisymmetric part of a square matrix and the ‘tilde operator’ is defined in eq. (1). 2.2 Masses The dynamic properties of a rigid body are uniquely described by its complete inertia tensor Ib . Consider a uniform sphere Bb and take a reference system Ψb fixed to the sphere and with its origin coincident to the origin of the sphere. For a general rigid body, the dynamics equations are: Pbb = Pbb ∧ Tbb,0 + Wbb tot
(3)
where Pbb = Ibb Tbb,0 ∈ 6 is the generalized momentum3 , Tbb,0 = ((ωbb,0 )T (vbb,0 )T )T is the twist of the sphere respect to an inertial frame Ψ0 , Wbb tot is the total wrench applied to the sphere and Pbb ∧ comes from the Lie-Poisson bracket and in the body coordinates Ψb is represented by a 6 × 6 matrix of the following form: b P˜ P˜vb (Pbb ∧) := ˜ωb Pv 0 T T T where Pbb = (P ω Pv ) . For the specific case of the jI3 0 sphere, Ibb = where I3 is the 3 × 3 identity 0 mI3 matrix, j is the rotational inertia of the sphere and m its mass.
2.3 Dampers The easiest manner to model a linear spatial damping effect is to use an element which generates a wrench directly proportional to the twist of the body whose free-energy has to be dissipated. In this paper we use: Wbb diss = RTbb,0
(4)
where R ∈ 6×6 is a positive definite matrix representing a dissipation tensor in the frame Ψb . 2.4 The proposed scheme In order to explain the proposed control scheme, and for the proper definition and execution of the manipulation tasks, a set of reference frames must be introduced, see Fig. 1. The fingers frames With reference to Fig. 1, let us suppose to have n fingers (or, more in general, limbs) which can be used to grasp an object. Let us define for each fingertip Bi a frame Ψi i = 1..n. Once a fixed reference frame Ψ0 has been assigned in space, we can represent Ψi with an element of SE(3), defining the change of coordinates Hi0 from Ψi to Ψ0 . We can therefore identify Ψi , as well as all the frames used in the following, with the matrix expressing the change of coordinate from the frame to be described to Ψ0 . It is useful to introduce a second frame Ψc(i) connected to the i-th tip which is called center of stiffness 3 The notation X j,k represents the motion of frame H respect i i to Hk expressed in the frame Hj .
Ψc(b) Ψv(1)
Ψn
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Ψ0
Figure 1: The basic frames. for the tip Bi and that is displaced from Ψi by a constant amount. The virtual object frames With reference to Sec. 2.2 and to Fig. 1, let us consider a virtual object of spherical form and uniform density to which we can associate a body frame Ψb , and an inertia tensor Ib . Displaced from Ψb and rigidly attached to it, we can consider a second frame Ψc(b) , which we call center of stiffness frame of the object. Following the previous c(b) ∈ SE(3) represents the notation, the element Hb change of coordinates from Ψb to Ψc(b) or the motion which brings Ψc(b) to Ψb . Furthermore, we have other n frames Ψb(i) i = 1..n rigidly attached to the virtual body, which all correspond to the application of a one-parameter group of b (si ) to the frame Ψb : suppose transformations Hb(i) b,b to have a twist Tb(i) associated to the finger i. We consider the frame Ψb(i) as the result of the action s T˜ b,b
s T˜ b,b
b of e i b(i) on Ψb (Hb(i) (si ) = e i b(i) ), where si ∈ is the parameter for the mentioned group of transforb,b ∈ se(3) is a properly chosen twist and mation, T˜b(i) e : se(3) → SE(3) is the surjective matrix exponential representing the exponential map of the Lie group SE(3):
b = Hb e Hb(i) = Hb Hb(i)
b,b si T˜b(i)
si
=e
0 b,b @ω˜ b(i) 0
1
b,b vb(i) A 0
As we explained with wrenches in Sec. 2.1, we can represent numerically twists as elements of se(3) in two different but equivalent ways: one possibility is by means of a 4 × 4 matrix as in the previous equation and the T b,b b,b b,b = (ωb(i) )T (vb(i) )T ∈ other one is as usual Tb(i) 6 . As last set of frames, we need a fixed displaced frame Ψv(i) , called the virtual frame for finger i, such that b(i) Ψv(i) is the result of the action of a fixed Hv(i) ∈ SE(3) to the frame Ψb(i) . The controller implements an elastic force which tends to align Ψc(i) with its correspondent c(i) Ψv(i) . The Hij used in eq. (2) is therefore Hv(i) or its inverse.
The object virtual frame In order to control the interaction between the grasped object and the environment, an extra frame Ψv(b) called the object virtual frame is needed. The controller implements an elastic force which tends to align Ψc(b) with Ψv(b) . 3 The general control scheme It can be shown [4], that the proposed control scheme can be represented as a generalized Hamiltonian system plus dissipation and therefore has nice passivity properties. The proposed controller is ‘physically’ passive in the sense that it does not create energy since the controlled system has a physical structure. By following Hogan’s physical equivalence principle [1], we control the robot in such a way that its behavior can be described by a mechanical system composed of springs, dampers and masses. We consider the virtual object as existent and as part of our system. This object, having a mass, contributes to the total energy of the controlled system. We then define n+ 1 3D springs which connect each Ψc(i) to its corresponding virtual frame Ψv(i) and one which connects the virtual object frame Ψv(b) to the object center of stiffness frame Ψc(b) . In order to damp out oscillations of the virtual object respect to the hand, we simulate in our controller a viscous force which dissipates free energy. This feature allows to inject damping in the system and gives a strictly passive behavior by means of only position measurements [10]. The dynamics of the virtual object is simulated in real time within the controller where its state and therefore its twist is clearly known. The structure of the control algorithm is shown in Fig. 2. It can be decomposed in a certain number of units, some of which could be implemented as concurrent processes to distribute computational load. 3.1 Simulation of the Virtual Object The major block is the Virtual Object Simulation block (see Fig. 2). This block simulates the dynamics of the virtual object in runtime using a proper integration algorithm. It has as inputs the sum of the wrenches as shown in Fig. 2. Integrating eq. (3) in real time, we obtain Pbb and can calculate Tbb,0 = (Ibb )−1 Pbb . It is then possible to prop-
Hb Wbb tot Virtual Object S Tbb,0 Simulation −Wbb diss
this way it is possible to inject damping in a physical way and without measuring velocities. By saying “in a physical way” we mean that the construction is equivalent to a physical system composed of spring masses and dampers and it is therefore passive and easily inb can be easily terpretable. The damping wrench Wdiss calculated using eq. (4) after choosing a damping matrix R. Notice that this approach is fundamentally different to approaches where either an observer or a state variable filter is used to have a velocity information which can be used to implement damping: here damping is injected directly in a physical way without the necessity to first estimate the velocities of the fingers5 . Furthermore, the virtual object can be used as a representative for the focus of the object to be grasp in some grasping strategies [4].
Rm H1
b −Wv(1),c(1)
Spring 1
Hb Hn
b −Wv(n),c(n)
Spring n
Hb
s = (s1 , . . . , sn ) b −Wc(b),v(b)
T −J¯tips ..
Hv(b)
Object Spring
τ
Robot
Hv(b)
Hb
q
Dir. Kin.
Figure 2: The simplified grasping control scheme. erly integrate the twist Tbb,0 in order to obtain Hb by using what is usually called Lie group left translation 4 : Hb (t + dt) = Hb (t) + H˙ b (t)dt = Hb (t) + (LH )∗ T˜ b,0 (t)dt = b
b
Hb (t) + Hb (t)T˜bb,0 (t)dt
(5)
3.2 The Injected Damping and the Virtual Object Concept In order to give the system a proper behavior, artificial damping is necessary. Usually, the implementation of damping requires measurements of velocities of the joints of the robot which are very often not available. Alternative schemes using the damping injection concepts [10] can be used instead as done in this work. The robot joints velocities are not known, but the twist of the virtual object is obviously available since it corresponds to a controller state. It is therefore possible to create a damping force, which is proportional to a twist, which acts on the virtual object. Since we attach the virtual object to the fingers by means of some elastic elements, the energy present in the fingers is transferred through these springs to the virtual object and there dissipated. In 4 It is clearly important in the integration method to implement a constrain that ensures that the sub-matrix Rb ∈ SO(3) of Hb ∈ SE(3) remains orthonormal during the integration process, in order for Hb to be representative of a proper isometry. This can be done by means of an orthogonalization procedure at each integration step or using other techniques like quaternions.
3.3 The Elastic Elements By following what done in Sec. 2.1, we can define appropriate springs between each of the tips and the virtual object and between the virtual object and its virtual frame Ψv(b) , by choosing, depending on the type x , with x = of task, proper relative positions for Hc(x) b 1..n and Hc(b) , and proper n + 1 spatial springs which tend to align Ψc(x) to Ψv(x) . The potential energy of the controller corresponds to the sum of the potential energies of these n + 1 springs. i the As shown in Sec. 2.1, we indicate with −Wi,j wrench generated by the spring connecting body i to body j applied on body i, where i, j = 1..n, b, v and body v indicates a virtual body position set by the user (Hv(b) ). The power continuous coupling of the controller and the robot, is implemented through energy ports represented by the interconnection of the controller elastic elements attached to the tips and the tips them self. If we indicate with J(q) the geometric Jacobian which maps joints velocities of the mechanism q˙ to the twists Tib,0 with i = 1..n of the tips respect to the inertial frame 0 and expressed in the virtual body frame b, we have that: b,0 b −Wc(1),v(1) T1 . .. T .. = J(q, Hb )q˙ ⇒ τ = J (q, Hb ) . Tnb,0
b −Wc(n),v(n)
b is the elastic wrench applied to the tip where −Wc(i),v(i) by the spring i expressed in frame b and it is opposite b to the wrench Wv(i),c(i) generated on the virtual body by the i elastic element due to the nodicity of a spring (see Fig. 2). 5 If the dynamics of the virtual object is simulated correctly, there are here no such problems as model mismatch as it would happen in a model based observer: the virtual object will always dissipate and ensure stability.
a) Joint position (K=1.5 digit/deg, Kc=2 digit/deg , b=0 digit sec) 50 40 30
(deg)
The chosen technique does also not create problems for the initialisation stage6 .
joint friction
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Figure 4: Stability of the control loop without (a) and with (b) damping. Virtual position θv and joint position θ 80 60
θv
40
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20
(deg)
4 Experimental Results A simple laboratory setup has been used for a first evaluation of the control technique described above. The setup, schematically shown in Fig. 3, consists of two one-dof “fingers” equipped with position, force/torque and tactile sensors, [11, 12]. The kinematics of the set-up is trivial, and the theory illustrated in the previous sections drastically simplifies to a one-dimensional case. The goal is to demonstrate with a simple setup the idea of the virtual object concept during manipulation. Obviously, generalizations to more complex 3D systems can be made using the theory presented in the previous Sections.
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Figure 5: Desired and real position of the finger.
Details on the implementation schema can be found in [13]. In the proposed schema only position measurement is used in the control loop (x1 , x2 ): force/torque and tactile information are acquired only in order to show the features and the performances of the overall control scheme. In the following, two types of results are presented. The first one refers to the damping injection and its stabilization features for the control loop, the second one illustrates the performances of the scheme in simple manipulation tasks.
Normal force: Kc=2.5 (solid) and Kc = 1 (dash) digit/deg, K=12 digit/deg 40
30
(N)
Figure 3: Experimental setup.
second one, when damping is injected (at t ≈ 4.5 s), oscillations are quickly stopped. Fig. 5 reports an experiment still involving only one finger. In this case, the desired motion θv (t) of the virtual object follows a sinusoidal trajectory (dashed curve), causing an unplanned contact with an obstacle present in the workspace. From the plot of the real position (solid curve) it may be seen that the freespace/contact transition does not cause any problem. The forces applied to the obstacle are reported in Fig. 6, considering two different values for the stiffness coefficient kc .
20
10
Fig. 4, Fig. 5 and Fig. 6 show some experiments concerning the damping injection. In particular, Fig. 4 reports an experiment in which an external disturbance is applied to one of the two fingers. The time-history of the joint position without (plot a) and with (plot b) the damping action is shown when it is switched on. Note that in the first case there is practically no damping (some energy is anyhow dissipated by the friction present in the motor and in the joint), while in the 6 As seen in Sec. 3.1, the controller is dynamical and therefore it has some initial conditions that have to be set. Precisely, the configuration of the virtual object and its twist have to be set in the initialization phase. One proper choice is a zero twist indicating that the virtual object is still and a position for it which is in the neighborhood of the center of the fingers. If the inertial parameters m and j of the virtual object are chosen small respect to the fingers inertial parameters, the virtual object will start moving and will come to rest in the position of minimal potential energy. It is therefore not necessary to be precise in the initial position of the virtual object because physics takes care of a proper initial state. The damping plays here an important role since it dissipates the eventual initial extra potential energy given to the system by not placing the virtual object in the position of minimal potential energy.
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Figure 6: Forces applied to the obstacle with two different values of kc .
The second type of experiment concerns the use of both the fingers for simple grasp and manipulation tasks of a spherical object. Fig. 7 reports data concerning a grasp of an object. The joint positions of the two fingers are shown in Fig. 7.a, along with the equilibrium length and the deformations of the virtual spring, which allow the motion and the object’s grasp. Fig. 7.b shows the time-history of the stored elastic energy and the force applied to the object, as measured by the force sensor. A result concerning the robustness of the control scheme is shown in Fig. 8. An external disturbance is applied to the object, plot (a), and the energy dissipated by the control is shown in plot (b). Note that, as already mentioned, the force sensor is not used in the control, but only to measure and report the forces applied during the grasp.
a) Joint position (θ1, θ2), equilibrium lenght (dashed) and deformation (dotted) of the variable springs 40
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b) Potential energy of the springs (grasp energy) and contact force (F ) n
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Figure 7: Result of a grasp task: (a) Joint positions, springs’ equilibrium length and deformations; (b) potential energies and contact force. a) External disturbance (Fn) 10 8
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b) Power dissipated by the damping injection 400
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Figure 8: External disturbance applied during the grasp (a) and relative dissipated energy for achieving stability (b).
5 Conclusions Basic features of the presented control strategy are the so-called virtual object concept and the damping injection principle. These features allow a design of the overall scheme based on very simple physical intuitions, as well as the definition in a natural manner of grasping and manipulation tasks. Experimental results, although at a preliminary stage, are very satisfactory and encouraging. Future work will deal with the implementation of this scheme on more sophisticated robotic hands for manipulation purposes, as well as on industrial robots for experimentation in interaction tasks. Acknowledgments. This work has been partially supported by MURST and ASI. Notation Ψi Right handed orthonormal coordinate frame i. Hij Homogeneous coordinate transformation from Ψi to Ψj . j Hi Homogeneous coordinate transformation from Ψi to Ψj . Tij Twist of Ψi with respect to Ψj . Tik,j Twist of Ψi with respect to Ψj as a numerical vector expressed in Ψk . Wi Wrench applied to a mass attached to Ψi . Wik Wrench applied to a mass attached to Ψi expressed as a numerical vector expressed in Ψk . Wi,j Wrench applied to a spring element connecting Ψi to Ψj on the side of Ψi . k Wi,j Wrench applied to a spring element connecting
Ψi to Ψj on the side of Ψi expressed as a numerical vector expressed in Ψk . Pi Momenta of body Bi . Pik Momenta of body Bi expressed as a numerical vector in Ψk . Ii Inertia tensor of body Bi . Iij Inertia tensor of body Bi expressed as a numerical vector in Ψj . References [1] Neville Hogan, “Impedance control: An approach to manipulation: Part I-Theory”, ASME J. of Dynamic Systems, Measurement and Control, vol. 107, pp. 1–7, March 1985. [2] Richard M. Murray, Zexiang Li, and S.Shankar Sastry, A Methematical Introduction to Robotic Manipulation, CRC Press, March 1994, ISBN 0-8493-7981-4. [3] Stefano Stramigioli, “A novel impedance grasping strategy based on the virtual object concept”, in Theory and Practice of Control and Systems, A.M. Perdon A. Tornanb`e, G.Conte, Ed. 1998, World Scientific, ISBN 981-023668-9. [4] Stefano Stramigioli, From Manifolds to Interactive Robot Control, PhD thesis, Delft University of Technology, Delft, The Netherlands, December 4 1998, ISBN 909011974-4, http://lcewww.et.tudelft.nl/~stramigi. [5] Josip Lonˇcari´c , Geometrical Analysis of Compliant Mechanisms in Robotics, PhD thesis, Harvard University, Cambridge (MA), April 3 1985. [6] Josip Lonˇcari´c , “Normal forms of stiffness and compliance matrices”, IEEE Trans. on Robotics and Automation, vol. 3, no. 6, pp. 567–572, December 1987. [7] Ernest D. Fasse, “On the spatial compliance of robotic manipulators”, ASME J.of Dynamic Systems, Measurement and Control, vol. 119, pp. 839–844, 1997. [8] Ernest D. Fasse and Peter C. Breedveld, “Modelling of elastically coupled bodies: Part i: General theory and geometric potential function method”, Accepted for publication in ASME J. of Dynamic Systems, Measurement and Control, 1997. [9] Ernest D. Fasse and Peter C. Breedveld, “Modelling of elastically coupled bodies: Part ii: Exponential- and generalized-coordinate methods”, Accepted for publications in ASME J. of Dynamic Systems, Measurement and Control, 1997. [10] Stefano Stramigioli, “Creating artificial damping by means of damping injection”, in Proceedings of the ASME Dynamic Systems and Control Division, K.Danai, Ed., Atlanta, (GE), 1996, vol. DSC.58, pp. 601–606. [11] A. Cicchetti, A. Eusebi, C. Melchiorri, and G. Vassura, “An intrinsic tactile force sensor for robotic manipulation”, in Proc. 7th. Int. Conf. on Advanced Robotics, ICAR’95, Sant Feliu de Guixols, Spain, 1995. [12] C. Melchiorri, “Translational and rotational slip detection and control in robotic manipulation”, in 14th IFAC World Congress, Beijing, China, July 5 - 9, 1999. [13] Claudio Melchiorri, Stefano Stramigioli, and Stefano Andreotti, “Using damping injection and passivity in robotic manipulation”, in Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Atlanta, USA, Dec. 1999.
