串联弹性执行器的阻抗控制：无源与加速度控制.pdf

发布时间：2022-06-08 发布人：admin 分类：说明书资料大小：2.31M 资料格式：pdf 举报版权申诉

qq_44657800-11898169-4744302543288453569.pdf-第1页.png

第1页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第2页.png

第2页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第3页.png

第3页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第4页.png

第4页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第5页.png

第5页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第6页.png

第6页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第7页.png

第7页 / 共12页

qq_44657800-11898169-4744302543288453569.pdf-第8页.png

第8页 / 共12页

Impedance control of series elastic actuators: Passivity and acceleration-based control

1 Introduction

2 Impedance control of series elastic actuators

2.1 Modeling

2.2 Notation

2.3 Impedance control architectures

2.4 Collocated admittance architecture

2.5 Collocated Impedance Control

3 Passivity of existing solutions

3.1 Basic Impedance Control (BIC)

3.2 Velocity-Sourced Impedance Control (VSIC)

3.3 Collocated Admittance Control (CAC)

3.4 Collocated Impedance Control (CIC)

4 Acceleration-based impedance control

5 Experimental validation

5.1 Experiments to assess the stability region

5.2 Experiments to assess the impedance accuracy

6 Conclusion

Appendix A

A1 Deriving condition (16)

A2 Deriving condition (21)

A3 Passivity of Collocated Impedance Control

References

Mechatronics 47 (2017) 37–48 Contents lists available at ScienceDirect Mechatronics journal homepage: www.elsevier.com/locate/mechatronics Impedance control of series elastic actuators: Passivity and R acceleration-based control ∗ Andrea Calanca , Riccardo Muradore , Paolo Fiorini Department of Computer Science, University of Verona, Strada Le Grazie 15, 37134 Verona, Italy a r t i c l e i n f o a b s t r a c t Article history: Received 6 April 2017 Revised 5 August 2017 Accepted 20 August 2017 Available online 8 September 2017 Keywords: Series elastic actuators Impedance control Passivity Acceleration-based control 1. Introduction Series elastic joints allow force and impedance controllers to be implemented on high torque and high power density motors. Several impedance controllers have been proposed whose stability is usually an- alyzed by means of passivity-based tools such as the Z -width characterization. This paper proposes an overview of existing impedance control solutions for series elastic joints and derives the passivity char- acterizations that are still missing in the literature, thus providing a complete and coherent overview of the existing solutions. Within this overview, we highlight the advantages of impedance control based on positive acceleration feedback showing improved stability robustness and impedance accuracy with respect to existing solutions. These advantages are theoretically motivated (considering ideal conditions) and experimentally validated. © 2017 Elsevier Ltd. All rights reserved. Series elastic actuators (SEA’s) are an emerging technology to achieve high ﬁdelity force control of high power density motors [1] . In fact, series compliance can dramatically improve explicit force control robustness [2,3] . Series elastic joints have been suc- cessfully applied to humanoid robots (e.g. NASA’s Valkyrie [4] , Vir- ginia Tech’s THOR [5] and IIT’s COMAN [6] ), quadrupeds (e.g. ETH’ s StarlETH [7] ), modern rehabilitation and assistive robotics [8– 11] and cooperative robots (e.g. RethinkRobotics Baxter [12] ). Most of these applications are based on impedance control and need to deliver forces with a high level of safety and accuracy. While safety is a primary need for robots that interact with humans or with un- structured environments, the demand for high accuracy is increas- ing only recently. As an example, this requirement is pushed by novel haptic interfaces which have been proposed in the last years. Among others, Basafa et al. designed a haptic laparoscopic device with three degrees of freedom actuated by SEA’s [13] ; Zinn et al. proposed a haptic interface with large workspace based on the Dis- tributed Macro-Mini concept where the macro actuation is given by SEA’s [14] ; Oblack et al. proposed a multi-purpose rehabilitation haptic device using series visco-elastic actuators [15] ; Parietti et al. R This paper was recommended for publication by Editor-in-Chief is Proof. Reza Moheimani. ∗ Corresponding author. E-mail addresses: andrea.calanca@univr.it , andrea.calanca@gmail.com (A. Calanca), riccardo.muradore@univr.it (R. Muradore), paolo.ﬁorini@univr.it (P. Fiorini). http://dx.doi.org/10.1016/j.mechatronics.2017.08.010 0957-4158/© 2017 Elsevier Ltd. All rights reserved. designed a haptic device with a series visco-elastic elements for very ﬁne force rendering, in the range of the human sensory accu- racy [16] . In these haptic devices, the choice of SEA’s has been usu- ally motivated by the decoupling effect of the series spring which allows to mask the motor inertia and allows the accurate rendering of even very low forces. Differently from traditional haptic inter- faces, existing impedance controllers for SEA’s make use of explicit force feedback, meaning that the force is explicitly measured and fed back to the control system. This is because implicit force con- trol (where the force is delivered in open loop by controlling the motor current) cannot mask the motor inertia and cannot damp the series spring oscillations. The control problem of physical interaction (with humans or with unstructured environments) involving explicit force feedback is considered a hard challenge in robotics. Most of the proposed solutions are based on the passivity interaction paradigm, which provides a high level of stability robustness [17] . In particular pas- sivity of the controlled robot is a suﬃcient and necessary condi- tion to guarantee a stable interaction with any passive environ- ment [18] including humans who are usually assumed as passive systems [19] . Consequently several passivity-based control (PBC) algorithms have been proposed to shape the output impedance or the output force of SEA’s. The ﬁrst passive force controller for SEA’s was proposed by Pratt and Williamson [1] . Their solution is based on acceleration feedback that forces the motor to have the same acceleration of the environment, thus compensating for load motion and leading to robust performance, i.e. predictable error dynamics. Quite surprising this important feature was not

38 A. Calanca et al. / Mechatronics 47 (2017) 37–48 explicitly highlighted neither in the original work nor in the fol- lowing literature which has been focused more on stability robust- ness rather than performance robustness. Another investigation on PBC of SEA’s has been conducted by Vallery et al. where a veloc- ity sourced impedance control schema was considered [20] . They found out that “SEA cannot display a higher pure stiffness than the spring stiffness if passivity is desired”. Thus the spring design cannot be arbitrarily compliant but must be tailored to the max- imum desired stiffness leading to a trade-off between force con- trol robustness (which requires compliance) and the maximum dis- playable stiffness. One of the outcomes of the research described in this paper is to show that the same limitation highlighted by Vallery et al. holds for several existing control architectures: admittance con- trol [21] , impedance control [22] and parallel force-position control [23] . The admittance control architecture was proposed by Pratt et al. to reduce the force control bandwidth requirement in high impedance rendering [21] . We will show that even this algorithm cannot passively render a stiffness higher than the stiffness of the physical spring. A similar outcome emerges by analyzing the pas- sive impedance control of the DLR lightweight arm where a paral- lel force-position architecture is implemented [23] . In conclusion, we will show that at the current state of the art there exist no passive algorithms that allow to overcome the physical spring stiffness. Further limitations arise when a virtual Voigt model impedance is desired, i.e. the parallel of a spring and a damper. For example an impedance controlled SEA with velocity controlled mo- tor cannot passively display a pure Voigt model [24] and we will show that the same result holds for other control architectures. Thus, the ﬁrst contribution of this work is to derive missing pas- sivity conditions for existing impedance control architectures pro- viding a coherent framework of passivity results. The second contribution is a novel algorithm that allows in the- ory to passively overcome the physical spring stiffness and to pas- sively display a pure Voigt model. This algorithm is inspired by the seminal work by Pratt and Williamson [1] who used load acceler- ation feedback to control the SEA output force and to cancel out the inﬂuence of load dynamics in force control. We will formally show that by taking advantage of load dynamics cancellation it is possible to passively render any passive impedance. The intuition behind this approach is that by exactly compensating for the load motion, load (or environment) uncertainties disappear. In partic- ular we refer to (perfect) load motion compensation as the ability to move the motor homokinetically with the load or the environ- ment. Thus load motion compensation is a way to obtain virtual backdrivability, i.e. to backdrive a non-backdrivable motor by con- trol. Indeed, the force to accelerate the motor is transferred to the motor input by control and it is used to compensate for the motor inertia. In the past, the concept of load motion compensation has been already discussed in the force control literature and a generic framework of solutions is presented in [25] . However, the effects on impedance rendering and passivity have never been analyzed. Other examples of SEA’s force control where the load dynamics is explicitly taken into account includes disturbance observer archi- tectures [4,26–28] , adaptive [29,30] , robust [31] and sliding-mode [32] controllers. However, none of these works investigates the ef- fect of load motion compensation on impedance rendering. We highlight that the terms “load” and “environment” can be often considered as equivalent: they both refer to the dynamics the SEA or the robot is in contact with. A typical case is physical human- robot interaction (pHRI) where the environment include or is iden- tiﬁed with the human. To address the issues described above, the paper is organized as follows. To introduce the reader, Section 2 summarizes exist- ing impedance control algorithms for SEA’s. Section 3 derives pas- sivity conditions that are currently missing in the literature and provides a summary and easy-to-compare view of passivity results. Section 4 proposes a novel impedance control algorithm based on positive acceleration feedback. Section 5 experimentally compares the existing algorithms to our solution from the point of view of stability robustness and impedance accuracy. Finally, conclusions are drawn in Section 6 . 2. Impedance control of series elastic actuators Impedance control aims at shaping the dynamical relation be- tween the actuator position (or velocity) and applied external forces. Impedance control can be implemented using an inner force loop and an outer position loop or using the dual conﬁguration: an inner position loop and an outer force loop. The latter case is usually called admittance control. In both architectures the desired impedance/admittance is implemented in the outer loop while the inner loop must be fast enough to have negligible dynamics [33] . ∈ L 2 e An impedance relation mapping the velocity u into the or equivalently an admittance relation mapping the ∈ L 2 e into the velocity y is passive if there exists a con- ∈ L 2 e force y ∈ force u L 2 e β such that stant ≥ β, T ∀ u ∈ L 2 e , ∀ ≥ 0 0 T dt (t ) (t ) u y (1) −β represents the initial system energy [18,34] . If we con- where sider that ( u , y ) is a ﬂow and effort pair we have that the system output energy will never be greater than its input energy, over an = arbitrary observation time starting from t 0. The passive interac- tion paradigm is based on the assumption that the environment is a passive system and on the fact that negative feedback connection of passive systems is still passive, thus stable. Passivity of impedance controllers depends on several factors: the control architecture, the inherent system dynamics and the de- sired impedance. In particular, given a control algorithm applied to a system, the set of impedance values that can be passively rendered is called Z -width [35] . Established results in PBC liter- ature show that the controlled system dynamics cannot be too far from the inherent system dynamics [36] . In particular, Colgate et al. found that the virtual inertia of a current controlled motor cannot be reduced beyond one half of its physical inertia, using non-collocated proportional force feedback [17] . Similarly, Vallery et al. showed that the virtual stiffness of a SEA cannot be increased over the physical spring stiffness, considering a velocity sourced impedance control architecture [20] . Hereafter, several impedance control schemas are reported and their passivity is analyzed. We will start by introducing the com- mon modeling and notation where linear and rotary quantities are regarded as equivalent. 2.1. Modeling The following SEA model is considered = (θ − q ) k τe (2) ¨θ = J − τe τm (3) τ e is the spring force exerted on the environment (or torque where θ is the motor position, q is the joint in the case of a rotary joint), τ m is the (current-controlled) motor input force (or position and torque). Fig. 1 shows an equivalent linear representation of the sys- tem, which translates torques into forces and angular positions into linear positions. For the sake of generality, in this paper linear and angular quantities are used interchangeably. The SEA parameters are the spring stiffness k and the motor inertia J . In the existing SEA-passivity literature, the friction dynamics is usually neglected

A. Calanca et al. / Mechatronics 47 (2017) 37–48 39 Fig. 1. A model of linear SEA interacting with an environment. Equivalent linear representation which translates torques into forces and angular positions into linear positions. Table 1 Acronyms. Acronyms Meaning SEA PBC VM ∼ VM BIC VSIC CAC CIC AB Series elastic actuator Passivity-based control Voigt model Approximated VM Basic Impedance Control Velocity-Sourced Impedance Control Collocated Admittance Control Collocated Impedance Control Acceleration-based Fig. 2. Basic Impedance Control (BIC) which uses an outer non-collocated position loop to shape the impedance and an inner force control loop. leading to a worst case condition: friction is dissipative and would help to meet passivity. The same choice is made here to ease the comparison with similar results in the literature while keeping the focus on the methodology. 2.2. Notation Abbreviations and notations are listed in Table 1 and in the fol- lowing points: • • • • • The position and force control blocks are named C P ( s ) and C F ( s ), respectively. The impedance and admittance controllers are named I ( s ) and A ( s ), respectively. The position reference is q r , which describes the equilibrium position of the desired impedance. The motor is modeled as a pure inertia 1/ Js 2 according to Eq. (3) . The environment block is marked with the letter E and its τe ). However, for ease of rep- , input–output port is the pair ( ˙ q resentation we considered the velocity integrator included in the environment block letting the position q be the environ- ment output. Equivalent diagrams can be drawn considering ˙ q as output. 2.3. Impedance control architectures Fig. 2 represents the basic arrangement for SEA impedance con- trol. The inner force controller C F ( s ) is fed by an outer loop I ( s ) that measures the load position and computes the force reference needed to obtain the desired impedance. For example, to render a Fig. 3. Velocity-Sourced Impedance Control (VSIC) which uses an outer non- collocated position loop to shape the impedance and an inner force control loop with a nested velocity controller. Fig. 4. Collocated Admittance Control (CAC). The position loop is closed on motor θ (collocated) instead of on joint position q . The outer force loop shapes position the impedance. Voigt model (VM) impedance, the outer controller should be sI(s ) = + k d d d s (4) and d d where k d are the desired (positive and constant) stiff- ness and damping coeﬃcients. The inner force loop may use a proportional-derivative action while the integral term is often avoided due to the passivity constraint [1] . We call this control ar- chitecture Basic Impedance Control (BIC). Examples of this schema are in [22,37] . A common improvement of this schema is to add an inner mo- tor velocity loop as shown in Fig. 3 . This solution is often called “velocity-sourced” and helps to deal with friction in the trans- mission system [38,39] . Moreover, passivity allows for an integral action within the force loop [20,24] . We name this architecture Velocity-Sourced Impedance Control (VSIC). 2.4. Collocated admittance architecture Fig. 4 shows an admittance control approach with a motor po- sition loop and an outer force feedback [21] . The position control θ is collocated since the sensor is located on the same rigid loop on body of the actuator (i.e. the encoder is on the rotor) [40] . A posi- tion control loop on q would be non-collocated and so more diﬃ- cult to stabilize [41] . To render a visco-elastic impedance, the force controller com- putes the position reference as θ m,re f = − A (s ) s τe , = A (s ) s − k d /k 1 + sd dm k d (5) and d dm where k d are the desired stiffness and motor-level damp- ing. Differently from classical admittance implementations [33] , the position feedback is closed on the motor position instead of on the joint position, thus avoiding the non-collocation issue. For this reason we refer to this schema as Collocated Admittance Control (CAC). Unfortunately this architecture cannot render a pure VM dy- namics because the “virtual” damping cannot be rendered at joint- level (as it would require a non-collocated measure) but only at the motor side. In the ideal case of an inﬁnitely fast inner control loop, the controlled actuator dynamics would be ˙ θ + = k d q −τe , d dm meaning that the virtual damper can only act at the motor level and not at the joint level, thus resulting in an approximated Voigt ∼ VM ). On the contrary, the BIC and the VSIC architectures model ( can display a joint level damping leading in ideal conditions (in- ﬁnitely fast inner loops) to the VM dynamics + = ˙ q k d q d d −τe . (6) (7)

40 A. Calanca et al. / Mechatronics 47 (2017) 37–48 Fig. 5. Collocated Impedance Control (CIC). The architecture refers to its energetic interpretation as explained in [23] . The main difference with respect to BIC is that motor position is fed back instead of joint position. 2.5. Collocated Impedance Control An alternative impedance architecture has been proposed for controlling the DLR lightweight arm [23] . This robot is modeled with ﬂexible joints because of harmonic drive dynamics. From the modeling point of view a ﬂexible joint is similar to a SEA except for the higher magnitude of elastic and damping parameters. Con- ceptually, the impedance controller of the DLR lightweight arm is based on a Collocated Impedance Control (CIC) architecture where an inner force loop is fed by an outer (collocated) motor position loop, as shown in Fig. 5 . Similarly to the CAC case, collocation pre- vents virtual joint-level damping. In particular the outer controller can be computed as sI(s ) = + k dm d dm s (8) is the desired motor-level damping and k dm where d dm = sired motor-level stiffness. By posing k dm set the desired joint-level stiffness leading, in ideal conditions, to the desired dynamics (6) where kk d −k d , it is possible to k is the de- = k d kk dm + . k dm k (9) = Consequently, the physical spring limit k d k can never be reached and can only be approached for very large (theoretically inﬁnite) values of k dm . 3. Passivity of existing solutions This Section analyzes the passivity of existing impedance con- trol architectures. In particular two kinds of desired impedance are considered: a pure spring dynamics ( sI(s ) ) and a paral- lel spring-damper dynamics ( sI(s ) ). A compact overview of passivity results will then be summarized in Table 2 . + k d sd d k d = = Considering a linear time invariant system with impedance Z ( s ) the passivity deﬁnition (1) is equivalent to the conditions (i) Z ( s ) is + stable and (ii) Re [ Z(iω)] . In the following subsections, the conditions (i) and (ii) are derived for each control architecture, , where Z ( s ) is the impedance seen at the environment port i.e. ∀ ≥ 0 −Z(s ) ˙ q [18,34] . (τe ) , ˙ q ∈ = τe ω R 3.1. Basic Impedance Control (BIC) ω D . Solving for Re[ Z ( i ≥ 0 [34] the fol- )] + P = iω where a lowing condition is derived: 2 J k + 1 − ω − k d 1 k + k d ω 2 J ≥ 0 . (13) ≤ k , meaning that a desired stiff- Thus, passivity is equivalent to k d ness higher than the physical spring stiffness is not allowed by passiv- ity . = + k d sd d the condition (13) be- In the VM case, where sI(s ) comes + k d D − k d 1 k ω 2 J − ω P 2 (D 2 − JP ) k 2 + P + d d ≤ k and P ≤ k D J 2 . Since D is usually small which is implied by k d to avoid noise ampliﬁcation, the force proportional gain should be limited as well. This implies a bandwidth limitation for the inner force controlled system . Interestingly the bandwidth limit depends on the actuator bandwidth, i.e., the location of its mechanical reso- nance. ≥ 0 (14) 3.2. Velocity-Sourced Impedance Control (VSIC) Suﬃcient passivity conditions for the VSIC schema have been and the following struc- reported in [20,24] considering sI(s ) ture for the force and motor velocity controllers k d = C F (s ) = + P f C V (s ) = + P v I f , s I v . s (15) Here a more general suﬃcient condition is reported, which is de- rived in Appendix A.1 : passivity of the VSIC architecture is guaran- teed if ≤ αk, k d + I f I v α ≥ 0 depends on velocity and force gains ≤ P f P v J P f P v where α = + 2 v P f I 2 v P f I + (I v P f ) I f P v < . 1 k (16) (17) In conclusion 1. passivity limits the maximum desired stiffness to (less than) the physical spring stiffness , 2. given a suﬃciently low value for the velocity and force proportional gains, P f , no upper limit exists for P v 3. pure integrators are allowed both within the velocity and force loops, with limited integral gains. The smaller the integral gains the more k d can approach k . Finally, it has been shown that in the case sI(s ) VSIC architecture is not passive [24] . = + k d sd d the For the BIC case a proportional-derivative force control law is 3.3. Collocated Admittance Control (CAC) considered = + sD P (s ) C F (10) where P and D are positive constants. A pure integrator would vi- olate the passivity of the inner force loop [1] . The impedance Z ( s ) at the environment port can be computed as (τe ) , ˙ q For the CAC case a passive position controller is considered (s ) C P = + sD P where P and D are positive constants. The impedance Z ( s ) at the environment port is −τe = + I(s ) (s ) Js C F 2 + + (s ) J s C F k 1 ˙ q (11) −τe = skI(s 2 + C F 2 + )(Js C F − (k )) (s (s )) − k d ) s [ sI(s )(Js which is a stable transfer function. In the case sI(s ) compute = Re [ Z(iω)] + (k d D ) d d P a − P Dk d + 2 + ω 2 2 D a ω + 2 d d 2 D = k d one can ω 2 DJ (12) A (s ) s has been substituted with where k d ≤ k , meaning again that a desired stiff- passivity is equivalent to k d ness higher than the physical spring stiffness is not allowed by pas- , sd dm a suﬃcient and sivity . In the ∼ VM case, where sI(s ) + k d . For the case sI(s ) = ˙ q − skI(s )] −k d /k sI(s ) 1 (18) (19) =

Table 2 Passivity of existing impedance controllers. A . C a l a n c a e t a l . / M e c h a t r o n i c s 4 7 ( 2 0 1 7 ) 3 7 – 4 8 4 1

42 A. Calanca et al. / Mechatronics 47 (2017) 37–48 needed to compress the spring, to reach perfect torque tracking. In fact, the SEA model (2) –(3) can be rearranged as + J ¨q + J ¨τe k = τe τm (23) Fig. 6. Impedance control based on acceleration feedback. This architecture is iden- tical to the BIC scheme in Fig. 2 except from the additional term u a . necessary condition for passivity is d dm P D − 1 J = ¯k 1 d dm ¯k − k d + D ≤ k < k d (20) where . This means that the desired stiffness may be bounded from below. Interestingly, suﬃcient conditions for passiv- ity are k ≤ k, k d P D < k d d dm (21) meaning that the position gain P is upper limited. This implies bandwidth limitations for the position controlled sub-system de- pending on the desired impedance . Condition (21) also indicates that a high derivative gain D helps to meet passivity. The proofs of (20) and (21) are missed in the literature and are reported here in Appendix A.2 . 3.4. Collocated Impedance Control (CIC) Passivity of the CIC architecture has been already investigated considering an energy shaping interpretation [42] . In particular, the motor kinetic energy is shaped via inner force feedback and the potential energy via outer position feedback, then, the total energy function is used as a candidate Lyapunov function. Considering our modeling, passivity is equivalent to positiveness of P , D , k dm and . According to (9) setting a motor-level stiffness implies that d dm the maximum desired stiffness cannot overcome the physical spring stiffness , i.e., k d ≤ k . Details are reported in Appendix A.3 . 4. Acceleration-based impedance control The previous Section showed that all the existing implementa- tions of impedance and admittance control are constrained to the same passivity limitation, i.e. it is not possible to overcome the stiff- ness of the physical spring. In addition, it is not always possible to render an exact VM impedance due to passivity constraints and/or architecture limitations. The only architecture that allows the pas- sive rendering of an exact VM impedance is the BIC algorithm, if the inequality (14) is veriﬁed. This Section proposes an impedance controller based on posi- tive acceleration feedback, which allows to overcome the passiv- 1 This algorithm is designed ity limitations of existing architectures. considering the BIC architecture and adding the following term to the motor input torque + (t ) J ¨q d − q ) τ is the output of the outer impedance con- where τ troller, as shown in Fig. 6 . The feed-forward term ( t ) in (22) ac- d counts for the desired force reference and delivers it directly on considers the motor acceleration the motor input. The term J ¨τ k = )(q r sI(s (t ) d (22) (t ) (t ) u a = + d τ ¨τ J (t ) d k 1 A preliminary version of this Section appeared as abstract in the International Symposium on Wearable Robotics, 18–21 October, 2016. La Granja, Segovia, Spain [43] . (t ) ¨τe accounts for motor acceleration to compress J where the term k the spring and is peculiar of series elastic systems. Finally, the term J ¨q in (22) provides the torque to accelerate the motor homoki- netically with the environment leading to compensate for the in- ﬂuence of the environment dynamics. We highlight that some cau- tion is needed when implementing the term J ¨q : overestimation of J leads to feedback inversion and to loose passivity. In real im- plementation a slight underestimation of J is suggested to avoid this issue as discussed later in this section. By considering u a it is possible to write + ) (s I(s ) 2 Js + 2 + (s ) J s k Then, the overall impedance seen at the environment port can be computed considering the Laplace transform of (22) 1 + 2 + J s k −τe = C F C F − ˙ q (24) (s ) . u a (t ) C F 1 1 = u a 2 + s J k 1 I(s ) + Js ˙ q ˙ q and by substituting (25) in (24) one ends up with −τe = I(s ) ˙ q (25) (26) meaning that the shaped actuator impedance is equal to the desired impedance . This leads to a novel result: the passive rendering of any passive impedance , which is formally stated in the following Propo- sition. Proposition 4.1. Given the series elastic actuator (2) –(3) and the control law τm )(sθ I(s ) (27) τe ) (s = + −C F + u a = u a 2 s + J 1 k sI(s ) + s 2 J q (28) with I ( s ) passive and C F ( s ) stable, then the controlled system is passive at the environment port (τe ) , ˙ q . Hence, the knowledge of ¨q allows to overcome the passivity con- straint of all the existing architectures . In fact, the proposed control law allows, theoretically, to render unlimited desired stiffness and unlimited desired damping. Moreover, Eq. (26) states that, unlike other architectures, the rendered impedance is equal to the desired impedance over the entire frequency spectrum. Of course such the- oretical expectations are too optimistic when compared to practical implementations where the acceleration signal is usually noisy and with limited bandwidth. In addition, it is known that “discrete con- trol, time delays, actuator and sensor limitations and unmodeled dynamics can compromise passivity, making the implementation of passive control on real systems extraordinarily challenging” [36] . However, Proposition 4.1 provides a theoretical insight to motivate the use of BIC endorsed with positive acceleration feedback and the next Section shows experimentally that such architecture out- performs the existing solutions in terms of both impedance accu- racy and robustness. Before to proceed with the experimental part, we highlight a practical issue with the acceleration-based (AB) law (27) and (28) . It has been reported in the literature that feeding back the acceleration term J ¨q may lead to feedback inversion and instability in the case the parameter J is over-estimated [1,25] . As mentioned before the term J ¨q provides the torque to accel- erate the motor homokinetically with the environment, however, if the parameter J is over-estimated the motor accelerates more than the environment possibly leading to instability. For this reason a (t ) (t )

A. Calanca et al. / Mechatronics 47 (2017) 37–48 43 Fig. 7. The SEA prototype used as testbed: M is the motor, T is a torque sensor θ and q (used for identiﬁcation), S is a torsional spring and the angular quantities are measured by encoders E 1 and E 2 , respectively. L is a metal frame used as inertial load in uncoupled conditions. reduction gain 0 of (27) and (28) leading to substitute (28) with k J 1 can be used in practical implementation < < = u a 2 s + J 1 k sI(s ) + 2 k J J s q. (29) Fortunately, passivity of control law (27) –(29) is still veriﬁed, as stated in the following proposition where a proportional-derivative form for the controller C F ( s ) is assumed. Within the proof one can observe that – differently from Proposition 4.1 – a small k J causes the rendered impedance to deviate from desired impedance. Thus, the gain k J should be tuned as high as possible while allowing sta- ble acceleration feedback. Fig. 8. Upper plot: comparison between velocity by numerical position differentia- tion (blue) and inter-period measurements (red). Lower plot: comparison between acceleration estimations. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) Proposition 4.2. Given the series elastic actuator (2) –(3) and the control law (27) –(29) with k J 1, I ( s ) passive and C F ( s ) as in (10) , . then the controlled system is passive at the environment port (τe ) , ˙ q < Proof. By substituting (29) in (24) the following impedance rela- tion can be found: (1 (s ) − k J + −τe = (30) I(s ) + 1 . ˙ q ) 2 Js 2 + J s k C F As the parallel interconnection of passive system is still passive and I ( s ) is passive we just need to verify passivity of the remaining term = ) (31) (s ) I e − k J (1 + + sD P 2 Js 2 + J s k 1 which represents the deviation from the desired impedance I ( s ). It can be veriﬁed that Re [ I e ( i 0 which holds because D 0 is equivalent to ω)] > 2 D > 0. ω > 5. Experimental validation This Section presents an experimental comparison of the de- scribed control architectures in terms of experimental coupled sta- bility and impedance accuracy. Experiments are conducted in a physical human-robot interaction scenario where a human can ex- ert forces on the impedance controlled joint of the SEA prototype shown in Fig. 7 . The prototype is composed of a geared DC motor M connected in series to a spring S and then to a metal frame L . Two optical encoders ( E 1 and E 2) are used to measure the motor and the load positions (with a resolution of 0.018 degrees) while the torque is measured using spring displacement. The torque sen- sor T is used only for identiﬁcation purposes. Velocities are ob- tained by measuring pulse inter-periods (using hardware timers available on the encoder electronics) which give a better approx- imation than using ﬁnite differences, as shown in Fig. 8 . Acceler- ations are obtained by numerical differentiation of velocities with Table 3 System parameters of the SEA testbed. Parameter Symbol Value Spring stiffness Torque constant Motor inertia Load inertia k k t J J load . 49 N m/rad 2 . 29 N m/A 2 2 . 0208 kg/m 0 2 . 0 0021 kg/m Table 4 Control tuning. BIC VSIC CAC CIC AB C F C V C F C P C F C F Gain Tuning P D P v I v P f I f P D P D P D k J 80 1.5 1 0.45 100 50 100 1 50 1 80 1.5 0.8 an output stage ﬁlter at 20 Hz . System parameters have been esti- mated using a procedure similar to the one described in [44] and are listed in Table 3 . The control system runs on a standard PC equipped with a quad-core processor running a real-time process with kernel-like priority. The control process runs at 3 kHz and communicates with the motor drive and the sensor electronics via Ethercat protocol at the same rate. Position and force controllers have been tuned in order to achieve the highest performance (maximum band- width and minimum static error) within the allowed passivity con- straint and without introducing noise into the system. Table 4

44 A. Calanca et al. / Mechatronics 47 (2017) 37–48 Fig. 9. Experimental results. Column 1 (from left): experimental stability region, yellow cells represent stable responses while light and dark green cells represent critical and unstable responses; red lines represents the theoretical passivity constraints. Columns 2–4: impedance accuracy of pure stiffness (cols 2 and 3) and pure damping (col 4) desired impedances (red lines). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

分享到：

赞收藏

资料库

串联弹性执行器的阻抗控制：无源与加速度控制.pdf

相关推荐

课程资源

热门标签

最新资料