On the Feedback Linearization of Robots with Variable Joint Stiffness

Transcription

1 8 IEEE International Conference on Robotics an Automation Pasaena, CA, USA, May 19-3, 8 On the Feeback Linearization of Robots with Variable Joint Stiffness G. Palli an C. Melchiorri Dipartimento i Elettronica, Informatica e Sistemistica Università i Bologna Via Risorgimento, 4136 Bologna, Italy {gianluca.palli, clauio.melchiorri}@unibo.it Abstract Physical human-robot interaction reuires the evelopment of safe an epenable robots. This involves the mechanical esign of lightweight an compliant manipulators an the efinition of motion control laws that allow to combine compliant behavior in reaction to possible collisions, while preserving accuracy an performance of rigi robots in free space. In this framework, great attention has been given to robots manipulators with relevant elasticity at the joints/transmissions. While the moeling an control of robots with elastic joints of finite but constant stiffness is a wellestablishe topic, few results are available for the case of robot structures with variable joint stiffness mostly limite to the 1-of case. We present here a basic control stuy for a general class of multi-of manipulators with variable joint stiffness, taking into account ifferent possible moalities for changing the joint stiffness on the fly by an aitional set of commans. It is shown that nonlinear control laws, base either on static or ynamic state feeback, are able to exactly linearize the closeloop euations an allow to simultaneously impose a esire behavior to the robot motion an to the joint stiffness in an ecouple way. Illustrative simulations results are presente. Inex Terms Robot Manipulators, Elastic Joints, Variable Stiffness, Feeback Linearization, Nonlinear Systems. I. INTRODUCTION One of the current challenges in robotics is the introuction of robots in the human environment an, as a conseuence, the human-robot interaction/cooperation. The motivations of the growing interest on these topics can be foun on the possibility of employing robots for human everyay activities, for operations in angerous environments or for the assistance to eler or hanicappe people. The main reuirements for the introuction of robots in the human environment are safety an epenability of the robotic system 1,. In 3 it has been shown how these reuirements exclue the use of stanar inustrial robots for the interaction with humans, because of the intrinsic limitations on the safety of these evices ue to the inertia of the links an to the magnitue of the torue that the actuators can apply. A new generation of lightweight robots has been evelope to overcome many of the limitations of stanar inustrial manipulators in terms of performance, portability an exterity 4, 5. In 6 it has been shown how these evices can ensure a safe interaction with an unknown environment an with humans. Anyway, these goals have been achieve with a significant increment of the cost of the overall robotic evice, ue to the use of composite materials an high-performance sensors an actuators /8/$5. 8 IEEE A. De Luca Dipartimento i Informatica e Sistemistica Università egli Stui i Roma La Sapienza Via Ariosto 5, 185 Roma, Italy eluca@is.uniroma1.it As an alternative way to make robots intrinsically safe, it has been shown in how the safety of robotic arms can be improve, besies maintaining a low level of inertia, introucing also an high compliance at the mechanical level both in the joints of the robot an in the interface between the robot an the environment. In orer to obtain also an aeuate level of both static an ynamic performances, the use of variable stiffness evices allows to satisfy all the reuirements for a safe an accurate interaction with humans an unknown environments. With the aim of verifying the effectiveness of this approach, even if limite to the single joint case, several variable stiffness evices, an in particular antagonistic actuate joints, have been evelope 3, 7, 8. In this paper, the feeback linearization problem of robotic manipulators with variable joint stiffness has been analyze. The feeback linearization of robot with elastic joints is a well-known problem, extensively treate by several researchers in its original formulation 9 11 or in the revise version in which the ynamic coupling between the joints an the links are consiere 1, 13. In 14, 15 the interaction of elastic joint robots with the external environment are analyze. In all these works, the joint stiffness has been consiere as constant parameters. More recently, some moifications to the classic problem have been propose, such as in 16 where the effects of joint amping on the solution of the feeback linearization problem are iscusse, or in 17 where the feeback linearization of antagonistic actuate robotic arms is carrie out. This paper aims to show how the full state linearization an simultaneous control of both the position an the stiffness of the joints can be achieve via static or ynamic feeback for the general ynamic moel of a robotic manipulator with variable joint stiffness. We suppose that the mechanical stiffness of the joint can be moulate by means of external control inputs. II. DYNAMIC MODEL OF ROBOTS WITH VARIABLE JOINT STIFFNESS Our starting point is the general ynamic moel of robot manipulators with n elastic joints of finite, but constant stiffness. The moel is compose by the ynamics of n rigi boies (n links an n actuators), couple through the elastic joints. Let R n an θ R n be, respectively, the generalize coorinates of the riven links an of the riving actuators. Uner the simplifying moeling assumption use in 9 (namely, in the contribution to the robot kinetic energy, the angular velocity of the motors is ue only to their own

2 spinning), the ynamic moel can be written as 9: M() + N(, )+K ( θ) = (1) B θ + K (θ ) =, () where M() is the inertia matrix of the robot links, vector N(, ) contains the centrifugal, Coriolis, an gravity forces, K = iag{k 1,..., k n } > is the joint stiffness matrix, B = iag{b 1,..., b n } is the inertia matrix of the actuators, an R n are the motor torues. Damping at the joints can be also inclue see, e.g., 16. In the following, we will also use the euivalent notation with matrix K ( θ) =Φk, (3) Φ=iag{( 1 θ 1 ), ( θ ),...,( n θ n )} (4) an vector k = k 1... k n T R n. In this paper, the joint stiffness matrix K in es. (1 ) will not be consiere constant but, in general, a function of time: K = K(t). (5) The range of values that joint stiffness can assume epens largely on the technological implementation of variable stiffness. In 7, joint stiffness can be varie in the range ; in 8, it can be varie from 13. up to more than 4; finally, in 3 the feasible range is (all units are Nm ra ). Without loss of generality, we assume that all joints will have variable stiffness. Moreover, it is K(t) > for all t, since it has no physical meaning to consier negative stiffness while if the stiffness rops to zero the joint/transmission woul lea to an unactuate system. Different causes may account for such stiffness variability. Joint stiffness can vary in response to extra comman inputs to the robot system, as a function of the system configuration, or from a combination of the two. Therefore, ifferent stiffness functions an behaviors can be efine epening on the system implementation. The simplest situation is when the joint stiffness k i can be irectly change by means of a (suitably scale) aitional comman ki,fori =1,...,n. In vector form, k = k. (6) This situation is consiere, e.g., in the paraigmatic 1-of case of two masses connecte through a spring of variable stiffness in 1,. Therefore, the overall available input u an the robot state x are: u = R n, x = k θ θ R4n. Inee, the ynamics of change of the stiffness parameters of the joints (or, more in general, of the transmissions) may not be neglecte. In this case, also in view of the mechanical nature of the system, we can moel the variation of joint stiffness as a secon-orer ynamic system of the general form k = φ(x, k, k, k ), (7) in which the epenence inclue also the stiffness an their time erivatives. In this case, euations (1 ) shoul be complemente by (7) in orer to represent the complete ynamic moel of a robot with variable joint stiffness. This moel covers a situation where the aitional actuation k moifies the motion transmission configuration so to change its stiffness, e.g., by pre-compressing nonlinear springs or by moving some mechanical parts. A slight generalization of this moel can be use also to escribe the case of antagonistic variable stiffness evices, such as those consiere in 3, 7, 8, 17 see the Appenix. As a result, the state vector of the robot is extene an becomes: x e = T T θ T θ T k T k T T R 6n, (8) so that e. (7) can be rewritten as: k = φ(x e, k ). (9) We note that for n =1, the number of state variable will be eual to that neee to escribe the ynamics of the variable stiffness actuating evice in 3. In all cases, the objective will be to simultaneously control the following set of outputs y = R k n, namely the link positions (an thus, through the robot irect kinematics, the en-effector pose) an the joint stiffness. The implications on feeback linearization an ecoupling control of the ifferent moels of joint stiffness variation will be investigate in Sec. IV. III. INVERSE DYNAMICS OF VARIABLE STIFFNESS ROBOTS For robot manipulators having elastic joints with variable stiffness, we consier here the problem of etermining the expression of the actuation commans an k, neee to perform an assigne motion task, with a preefine an simultaneous variation of the stiffness at the joints. These commans can be then use for efining the feeforwar action in a control scheme. We assume that the motion is specifie in terms of a esire smooth trajectory = (t) for the link variables (possibly coming from the kinematic inversion of a Cartesian trajectory) an that a esire time evolution of the joint stiffness matrix K = K (t) (or, euivalently, of the vector k = k (t)) is also given. These reference output trajectories may be the outcome of some optimization process, like the safe brachistochrone solution in. For illustration, the proceure is etaile using the simple moel (6) for joint stiffness actuation. Therefore, we have simply k, = k (t) an only the computation of the nominal motor torue is of actual interest. 1754

3 Note first that we can ifferentiate e. (1) twice with respect to time, without introucing erivatives of the input torue. We have thus M() 3 + Ṁ() + Ṅ(, )+ K ( θ)+k ( θ) =, (1) an M() 4 +Ṁ() 3 + M() + N(, )+ + K ( θ)+ K ( θ)+ K ( θ) =. (11) From e. (1), evaluate along the esire output trajectory, we obtain the reference motion for the motor position: θ = + K (M( ) + N(, )). (1) Evaluating now (1) along the esire trajectory, an using e. (1) for eliminating the presence of θ, yiels for the reference motor velocity θ = + K ( M( ) 3 + Ṁ( ) + Ṅ(, ) K ) K (M( ) + N(, )). (13) In the final step, by solving e. () with respect to θ θ = B K(θ ), (14) an substituting it into e. (11), we obtain an expression involving the motor torue, which appears pre-multiplie by matrix KB that is always non-singular. Therefore, we can evaluate this expression along the esire trajectory an solve for the reference motor torue as: = M( ) + N(, )+BK α with α = M( ) 4 +Ṁ( ) 3 +( M( )+K ) + N(, ) K K (M( ) 3 ( K K + K K (,,, 3,4 + Ṁ( ) + N(, )) ) (15) K ) (M( ) + N(, )) (16) where again es. (1) an (13) have been use to eliminate the explicit epenence on θ an θ. In the above expressions, the contribution ue to the rigi robot ynamics, to the joint elasticity, an to the time-varying nature of the latter can be clearly recognize. As a result of this analysis, some minimal smoothness reuirements are impose on the esire link motion (t) an on the esire stiffness profile K (t) in orer to achieve their exact reproucibility on a time interval,t of interest by the application of (15). For this, it is in fact necessary that (t) C 4 an k (t) C. The consiere moel oes not take into account issipative effects or har nonlinearities. If one wishes to inclue these in the inverse ynamics computation, if shoul be note that the link ynamics (1) nees to be ifferentiate twice whereas the motor ynamics () is never ifferentiate. Therefore, iscontinuous moels of friction or actuator eazones on the motor sie can be consiere without problems. On the other han, any such iscontinuous phenomena acting on the link sie shoul be approximate by a smooth moel. Furthermore, in the presence of actuator saturations, it is possible to keep the comman torues within the saturation limits by a suitable time scaling of the manipulator trajectory 18. IV. FEEDBACK LINEARIZATION OF ROBOTS WITH VARIABLE JOINT STIFFNESS The analysis on the feeback linearization control 19 of robots with joints of constant elasticity carrie out in previous works 9, 1, 16 is consiere as a starting point to evelop a general approach to the solution of the feeback linearization problem for robots with variable joint stiffness. It is necessary to assume that the joint stiffness are measurable uantities. This last assumption is not restrictive because, if practical implementations of variable stiffness evices are consiere 3, 7, 8, the joint stiffness is irectly relate to the system state variables. Then, the knowlege of the system state allows to compute the joint stiffness. Preliminarly, we note that feeback linearization is certainly not the simplest control strategy for the consiere system. However, it allows to prescribe linear tracking performance with arbitrary ynamics an to obtain exact trajectory reprouction in the nominal case. A. Static Feeback Linearization In this section, we assume that the joint stiffness epen on the (relative) positions of both the joints an the actuators 17. We suppose also that there is no coupling between the stiffness of the joints or, in other wors, that the stiffness of the i-th joint is influence only by the position of the joint itself, by the position of the i-th actuator an by the input ki that is use to moulate the stiffness. It is then possible to write k as generic nonlinear functions of the system state variables an θ: k i = β i ( i,θ i )+γ i ( i,θ i ) ki, i =1,..., n or, in a more compact form: k = β(, θ)+γ(, θ) k (17) where k is the vector of the joint stiffness. Note that this last euation is the form of e. (9), then all the consieration mae in the previous section for this stiffness moel are vali. E. (17), together with e. (1) an (), an recalling the efinition of K = iag{k 1,..., k n }, efine the complete moel of the robotic manipulator with configuration epenent joint stiffness. The stiffness k i has been written as a secon orer ifferential euation (see e. (17)) to highlight the fact that this moel represent the ynamics of a mechanism, actuate 1755

4 by the input ki, that moifies the configuration of the joint to changes its stiffness, i.e. by pre-compressing nonlinear springs or by moving some mechanical parts. From e. (17) it is possible to see that the vector relative egree of the output k i is two, while for the output, e. (1) remains unchange an, by consiering also e. (17), e. (11) can be rewritten as: M 4 +Ṁ3 + M + N + K ( B K(θ ) ) + K ( θ)+φ(β + γ k )= (18) where both the inputs an k appears, so we can conclue thatthevectorrelativeegreesof is four. Then, by recalling the state efinition (8), we can state that the conition for the solution of the feeback linearization problem on the sum of the relative egrees of the output information is satisfie, since the vector state imension is 6 while the vector relative egrees of an k are 4 an respectively. Then, the overall system can be written in a more compact form: where 4 k = α(xe ) β(, θ) + Q(x e ) k α(x e )= M Ṁ3 +( M + K) + N +KB K (θ )+ K ( θ)+φβ (19) () an Q(x e ) is the so calle ecoupling matrix: M Q(x e )= KB M Φ γ(, θ) (1) n n γ(, θ) To achieve non interacting control of both the positions an the stiffness of the joints of the robot, the ecoupling matrix Q(x e ) must be non singular. From e. (1) it is possible to see that this last conition is satisfie if both the iagonal matrices K an γ(, θ) are non singular, or in other wors: } k i > i =1,..., n () γ i ( i,θ i ) If these conitions are satisfie, by efining the static control law: ( ) = Q α(xe ) v (x e ) + (3) k β(, θ) v k we obtain the full linearize form of the overall system: 4 v = k v k where v an v k are the new inputs of the linearize system use to control, respectively, the positions an the stiffness of the joints of the manipulator. Note that the linearization law (3) is an algebraic function of the state x e an of the auxiliary inputs v an v k.to highlight this fact, the expression α(x e ) in e. () can be rewritten by substituting an 3 from e. (1) an (1). Another important remark is that iscontinuous phenomena in the motor ynamics (), e.g., ue to ry friction, as well as general nonlinearities in the joint stiffness ynamics (17) can be hanle within this control esign, since no ifferentiation of these relations is reuire. B. Dynamic Feeback Linearization By taking into account the following very simple stiffness variation moel: k i = ki (4) the full state linearization problem cannot be solve by means of a static state feeback, because the vector relative egree of the stiffness output becomes zero, an then the conition on the sum of the vector relative egrees of the output information is not longer satisfie. By rewriting e. (1) an (4), recalling the efinition (4), in the following form: M = N n n M + Φ k n n n n I n n k (5) it is possible to see that the ecoupling matrix of the system is singular, so it is not possible to achieve neither the noninteracting control an the full state linearization of this system via static state feeback. From the structure of the ecoupling matrix, one can see that a ynamic extension on the input k is neee to satisfy the conition on the vector relative egree of the outputs. Then, we efine the auxiliary control input u k by aing a chain of two integrators on the input k (see also Fig. 1): k = u k In this way the vector relative egree of k is, while, for what concerns, by ifferentiating e. (1) twice with respect to time we can write: M 4 +Ṁ3 + M + N + K ( B K(θ ) ) + Φ k +Φu k = (6) in which both the input an the input u k appear. This allows to state that the vector relative egree of is 4. By substituting k an k with k an k respectively, the system can be then rewritten as: 4 α(xe ) = + Q(x k e ) (7) n n uk where M Q(x e ) = KB M Φ (8) n n I n n α(x e ) = M Ṁ3 +( M + K) + N + Φ k + KB K (θ ), (9) from which it follows that the ecoupling matrix is nonsingular if K is non-singular, or, in other wors, if the joint stiffness are strictly positive. This conition has been alreay consiere before to maintain the physical meaning of the ynamic moel of the manipulator. 1756

5 Setpoint Generator v f z + z P + + Ψ v c Feeback Linearization Dynamic Extension u k k k Robotic Manipulator x e T k T θt θt T Fig. 1. Scheme of the feeback linearization controller with ynamic extension. Since the state, an then the imension of state space, is not change with respect to the previous analysis, it follows that the conitions for the solution of the full state linearization an non-interacting control problem on both the relative egrees of the outputs an the non-singularity of the ecoupling matrix are now satisfie. By efining the control law: ( = Q α(xe ) (x e ) + uk n n we obtain the linearize form of the (7): 4 v = k v k v v k ) (3) Note that, also in this case, the linearization law (3) is an algebraic function of the state x e an of the auxiliary inputs v an v k. To highlight this fact, the expression α(x e ) in e. (9) can be rewritten by substituting an 3 from e. (1) an (1). With respect to the previous case, the state linearization has been now achieve by means of ynamic feeback because of the introuction of the ouble integrator (ynamic extension) on the stiffness control input k. C. Control Strategy The state feeback linearization efine in the previous sections allows to control both the positions an the stiffness of the joint of the robot by means of two totally inepenent linear controllers, compose by a static state feeback plus feeforwar action: 3 v = 4 + P i ( i i ) (31) i= v k = k + P k1 ( k k)+p k (k k) (3) with iagonal gain matrices P k1,p k,p i,i=,...,3 such that: λ 4 + λ 3 p 3j + λ p j + λp 1j + p j = (33) λ + λp k1j + p kj = (34) with j =1,...,n, are Hurwitz polynomials, where p ij an p kij are the j-th term of the main iagonal of the gain matrix P i an P ki respectively, while i,i =,...,4 are the vector of the esire joint position an their time erivative up to the 4-th orer an k, k an k are the vector of the esire stiffness trajectories an their time erivative up to the -th orer. By means of these linear controllers, it is possible to achieve the asymptotic tracking of the position an stiffness trajectories if (t) C 4 an k (t) C. This control approach can be viewe as a static state feeback in the state space of the linearize system. E. (31) an (3) can be rewritten in a more compact form by grouping the control signals an the esire position an stiffness trajectories in this convenient way: z = v c = v v k, v f = T T T 3T 4 k To this en, it is useful also to efine the state vector z of the linearize system an the nonlinear coorinate transformation between the state of the original system an the state of the linearize system: h i T z = T T T 3T k T kt =Ψ(xe) = 6 4 k T k T T M N +Φk M h Ṁ M N +Φk+Ṅ +Φ k + Φ k k k It is important to note that, also in this case, both the state linearization an the outer linear control loop epens only on the state information x e an any time erivative of the outputs must be compute. The controller (31), (3) can be then rewritten as: where P = v c = v f + P z z =v f + P z Ψ(x e ) (35) i P P 1 P P 3 n n n n n n n n n n n n P k P k1 A scheme of the propose controller is epicte in Fig. 1. V. SIMULATION OF A TWO-LINK PLANAR MANIPULATOR The valiity of the propose approach is now reporte by presenting the simulation results of a planar two-link robotic arm with variable joint stiffness. Due to space limitations, the well-known ynamic moel of the arm an the solution of the previous euations for this system are omitte. Only the simulation results in the case of full state linearization via ynamic feeback escribe in Sec. IV-B are reporte, 1757

6 Description Symbol Value Joint inertia J j 1.15e- kg m Joint viscous friction coeff. j.1 N s m Joint mass m j.541 kg Link center of mass l c.85 m Link length l.3 m Motors inertia b m 6.6e-5 kg m Motors viscous frict. coeff. m.46 N s m Position error weight w 1 6 ra Stiffness error weight w k 1 4 ra N m Actuator torue weight w 1 N m TABLE I PARAMETERS OF THE -LINK PLANAR MANIPULATOR. because this case is more complicate from the implementation point of view with respect to the static one, an because there are no significant ifference in the response of the full state linearization via static or ynamic feeback. In the simulation scheme, the trajectories are generate through proper filters to compute also their erivatives up to the appropriate orer. The control strategy has been chosen as in e. (35) an the matrix P is obtaine from the solution of the CARE 1 euation with a iagonal state weights matrix. The parameters of the -link planar manipulator an of the controller use in simulation are reporte in Tab. I. The links of the manipulator are consiere ientical for simplicity. The elements of the resulting feeback matrix P are all iagonal: P ii = 316.3,P 1ii = 111.9,P ii = 19. P 3ii =19.6,P kii = 316.,P k1ii =5.1 In Fig. the positions an the stiffness of the joints of the two-link planar manipulator are reporte, together with their trajectory tracking errors. Note that the tracking errors are practically zero, as expecte. Both step an sinusoial joint trajectories are use together with coorinate movements to show the stabilizing properties of the controller. It is important to note that the joint stiffness trajectories are not affecte by the changes of the joint positions an vice versa. VI. CONCLUSIONS In this paper, the feeforwar control action neee to perform a esire motion profile on a robotic manipulator with joint variable stiffness has been compute an the problem of feeback linearization of these evices has been analyze. The key point in the analysis of this problem is the efinition of the stiffness moel, an in particular of the way the inputs of the system act to moulate the stiffness of the joints. Two cases has been consiere, with ifferent (vector) relative egree of both the position an the stiffness information. The results of this analysis are summarize in Tab. II. If also the amping of the transmission system is consiere, only input-output linearization can be achieve 16. The simultaneous non-interactive stiffness-position control can be implemente by means of an outer linear control loop, that can be seen as a static state feeback in the state space of the linearize system. The asymptotic trajectory tracking problem can then be solve with arbitrary ynamics if the 1 Continuous Algebraic Riccati Euation 1 Pos Joint 1 Pos Joint Joint positions ra.5 1 x Err Joint 1 Err Joint Stiff Joint 1.5 Stiff Joint x 1 5 Position errors ra Joint Stiffness N mra Stiffness errors N mra Time s Time s Err Joint 1 Err Joint Fig.. Full state linearization via ynamic feeback: (a) Joint positions, (b) position errors, (c) joint stiffness an () stiffness errors. position an the stiffness trajectories are continuous together with their time erivatives up to the 4th an n orer respectively. The experimental valiation of the propose approach will be consiere in the next future. The mixe case, in which both rigi an elastic joints are present, with constant or variable stiffness, can be easily solve on the base of the analysis reporte in this paper. Also the case of joints with ifferent stiffness variation moels can be easily consiere. Type of nonlinear state feeback Static Dynamic k i = β +γ ki full linearization not neee Stiffness moel k i = ki not enough full linearization Dimension n TABLE II SUMMARY OF THE RESULTS ON THE FEEDBACK LINEARIZATION. Throughout the paper, we have assume ieal conitions, a perfect knowlege of the robot ynamics moels an the availability of full state measures. The presente results can be use as a starting point for the efinition of aaptive an robust controllers for robots with variable joint stiffness, possibly base only on the feeback of the position information of both the links an the actuators, supposing the velocity information not available. APPENDIX ANTAGONISTIC VARIABLE STIFFNESS DEVICES In antagonistic variable stiffness evices 3, 7, 8, 17, a couple of actuators for each joint is present, an both these actuators contribute to etermine the position an the stiffness of the joint to which the actuators are connecte. The ynamic moel of the whole manipulator can be written by grouping the actuators in two sets, enominate here as α an β, escribe by two euations similar to (), one for each actuators set: M() + N(, )+η α η β = (36) B θ α + η α = α (37) B θ β + η β = β (38) 1758

7 where η αi = η αi ( i,θ αi θ βi ) an η βi = η βi ( i,θ αi θ βi ) represent the coupling torues between the i-th joint of the robot an its actuators, while α an β are the torues applie by the actuator set α an β respectively. By making the moel of elastic joint robots more general, e.g., by means of a suitable change of coorinates, it is possible to write the ynamic moel of antagonistic variable stiffness evices in a form similar to e. (1), () an (9). This generalization is meaningful since it allows to apply the general concepts presente in this paper to ifferent technological implementations of variable stiffness evices. By introucing the auxiliary variables p = θα θ β an s = θ α + θ β representing the positions of the generalize joint actuators an the state of the virtual stiffness actuators respectively, it is possible to write: M() + N(, )+F (s)g( p) = (39) B p + F (s)g(p ) = (4) B s + h( p, s) = k (41) where = α β an k = α + β, F (s) is a iagonal matrix whose elements are strictly positive functions representing the generalize joint stiffness, g( p) is a vector whose elements are o strictly monotonically increasing functions representing the generalize joint isplacements, while h( p, s) is a vector whose elements are functions such that h i (, ) =. As case stuies, for the antagonistic actuate robot escribe in 8, 17, in which transmission elements with exponential force/compression characteristic are use, the following relations hol: f i (s i ) = e asi g i ( i p i ) = b sinh ( c ( i p i ) ) h i ( i p i,s i ) = cosh ( c ( i p i ) ) e asi 1 where a, b, c, an are suitable constants an f i (s i ) are the elements of the iagonal of the matrix F (s). For the same variable stiffness evice, if transmission elements with uaratic force/compression characteristic are consiere 7, 17, the following relations hol: f i (s i ) = a 1 s i + a g i ( i p i ) = i p i h i ( i p i,s i ) = b 1 s i + b ( i p i ) where a 1, a, b 1,anb are suitable constants an f i (s i ) are the elements of the iagonal of the matrix F (s). For the variable stiffness actuation joint (VSA), escribe in 3, the thir-orer polynomial approximation of the transmission moel reporte in can be use to transform the system in the esire form: f i (s i ) = a 1 s i + a s i + a 3 g i ( i p i ) = i p i h i ( i p i,s i ) = b 1 s 3 i + b ( i p i ) s i + b 3 s i where a i,b i,i=1,...,3 are suitable constants an f i (s i ) are the elements of the iagonal of the matrix F (s). ACKNOWLEDGMENTS This work was partly supporte by the PHRIENDS Project, fune uner the 6th FP of the European Community uner STREP Contract IST REFERENCES 1 A. Bicchi, S. L. Rizzini, an G. Tonietti, Compliant esign for intrinsic safety: General issue an preliminary esign, in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots an Systems, 1, pp A. Bicchi an G. Tonietti, Fast an soft arm tactics: Dealing with the safety-performance trae-off in robot arms esign an control, IEEE Robotics an Automation Magazine, vol. 11, no., pp. 33, 4. 3 G. Tonietti, R. Schiavi, an A. 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