II. MODELLING. The concept of the Actuator with Adjustable Stiffness. (AwAS) relies on a lever mechanism. The variation of the joint stiffness is obtained through ...
2012 IEEE International Conference on Robotics and Automation RiverCentre, Saint Paul, Minnesota, USA May 14-18, 2012
A Position and Stiffness Control Strategy for Variable Stiffness Actuators I. Sardellitti, G. Medrano-Cerda, N. G. Tsagarakis, A. Jafari, D. G. Caldwell Istituto Italiano di Tecnologia (IIT)
Abstract— Variable stiffness actuators (VSAs) have been introduced to improve, at the design level, the safety and the energy efficiency of the new generation of robots that have to interact closely with humans. A wide variety of design solutions have recently been proposed, and a common factor in most of the VSAs is the introduction of a flexible transmission with varying stiffness. This, from the control perspective, usually implies a nonlinear actuation plant with varying dynamics following time-varying parameters, which requires more complex control strategies with respect to those developed for flexible joints with a constant stiffness. For this reason, this paper proposes an approach for controlling the link position and stiffness of a VSA. The link positioning relies on a LQR-based gain scheduling approach useful for continuously adjusting the control effort based on the current stiffness of the flexible transmission. The stiffness perceived at the output link is adjusted to match the varying task requirements through the combination of the positioning gains and the mechanical stiffness. The stability of the overall strategy is briefly discussed. The effectiveness of the controller in terms of tracking performance and stiffness adjustment is verified through experiments on the Actuator with Adjustable Stiffness (AwAS).
I. I NTRODUCTION Numerous Variable Stiffness Actuators (VSAs) have been recently developed by the robotics community as a valuable approach to tackle the problem of safety [1] and energy efficiency [2] in the new generation of robots. The main feature of these actuators consists in their varying dynamics through an adjustable stiffness mechanism, useful to cope with the common variability of a task. Several mechanical arrangements have been recently presented in literature, e.g. the antagonistic configuration, with a pair of actuators coupled through nonlinear springs in series [3]–[5], or the serial configuration with two independent actuators for the joint positioning and the stiffness adjustment [6]–[10]. From a control perspective, the variation of the stiffness introduces physical modifications in the actuation, requiring the control system to be subjected to quick and large transitions among different operating conditions. The effects of the variable flexibility on the overall dynamic performance are usually very significant and they are sources of trajectory tracking errors and undesirable vibrations. In addition, coupling phenomena between stiffness and positioning mechanisms which arise in most of the developed VSAs, require the introduction of nonlinear control techniques or linearization methods. The authors are with the Advanced Robotics Dept., Italian Institute of Technology, Genoa, Italy. Email: {irene.sardellitti,
gustavo.cerda, nikos.tsagarakis, amir.jafari, darwin.caldwell}@iit.it 978-1-4673-1405-3/12/$31.00 ©2012 IEEE
Significant research has been carried out on the control of the VSAs. Earlier strategy proposed in literature consisted in a PD-based controller designed on the linearized actuation plant [11]. Following this, nonlinear control techniques were explored such as the feedback linearization. This strategy, mostly implemented for flexible joints, was extended to control the VSAs in [12] as well as in [13]. However, a control strategy with fixed gains may result limited in performance when the dynamic variations of the VSA become significant. Alternative approach is the gain scheduling, which adjusts the controller action on the basis of a parameter that significantly affects the linearized dynamic variations of the system [14]. This method was implemented in [15], to control the linearized dynamics of the VSA. In contrast with the feedback linearization, this strategy uses the physical variables in the plant model, without introducing coordinate transformations. In this paper we improved the performance of the gain scheduling strategy in the link positioning of a VSA, despite the significant parametric variations of the actuation’s plant. For this purpose, the gain scheduling was designed through polynomial fitting of the gains associated with a set of Linear Quadratic Regulators (LQRs), and the stiffness level at the transmission was exploited as the scheduling variable. In particular, the LQRs were designed for a set of Linear Time Invariant systems arising from the Linear Parameter Varying approximation of the original nonlinear actuation system dynamics. Among the different control techniques, the main merits of the LQR are that it allows for actively damping undesirable vibrations which arise from the flexible transmission [16] and it ensures local stability around the operating point through full state feedback [17]. Following this, we addressed the control of the stiffness perceived at the output link by exploiting the gain scheduling. This allowed for expressing the desired stiffness behavior at the link in terms of elastic transmission reference, taking into account the effects of control gains for positioning. Therefore, we derived the reference for the elastic transmission which was controlled through a PID action. The effectiveness of the overall approach was tested on the Actuator with Adjustable Stiffness (AwAS) [18]. The paper is organized as follows. We first provide the model of the AwAS. Then, in Section III, the problem statement is discussed. Section IV describes the position control strategy which is followed in Section V by the description of the stiffness controller. Section VI briefly discusses about the stability. Section VII is focused on the experimental results.
2785
In Section VIII there are the conclusions of the work. II. M ODELLING The concept of the Actuator with Adjustable Stiffness (AwAS) relies on a lever mechanism. The variation of the joint stiffness is obtained through adjustment of the relative distance between the springs and the center of rotation of the joint (Fig. 1). Details about the mechanisms are presented in [18]. Given the lever arm r, which is the effective distance
c
with φi = νi2 Di +νi2 Kti Kei /Rmi , µi = −νi Kti /Rmi and ui the input voltage with i ∈ [1, 2]. Note that the negative signs in µi are introduced to take into account the polarity of the motors. The external torque applied at the link is represented ∂E with τext , and τr = ∂θ is the torque applied at the motor 2 M2 given by τr = −2ks nr sin2 (θs ). (6) The nominal values in (5) are presented in Table I. Note that, to simplify the notation the motor inertias (Bi ) are already scaled by the transmission ratios.
e
φ2
θ2 B2
b
φ1 d
a
f
θ1
q
I
B1 σ
Fig. 1. AwAS drawing. The motor M1 (a) rotates the intermediate link (b) around the joint axis of rotation (c). The motor M2 (d) drives a ballscrew mechanism moving a slider equipped with two antagonistic springs (e). The output link (f) is connected to the intermediate link (b) through the springs (e).
Fig. 2.
between the center of rotation of the joint (c) and the springs (e), the elastic potential energy E stored in the elastic mechanism is given by E = ks r2 sin2 (θs )
Description Link Inertia Link Damping Motor + Gearbox Inertia Motor Damping Gear ratio Torque constant Back-EMF constant Motor Resistance Motor + Gearbox Inertia Motor Damping Gear ratio Torque constant Back-EMF constant Motor Resistance Equivalent spring Transmission ratio Length lever arm (max)
where ks is the spring rate associated with the two antagonistic springs, and θs = q−θ1 is the spring deflection with q and θ1 being the generalized coordinates of the link and motor ∂E and (M1 ) position, respectively. The elastic torque τE = ∂θ s ∂τE the stiffness σ = ∂θs are therefore obtained as (2)
σ = 2ks r2 cos(2θs ).
(3)
Given the spring rate ks , the stiffness in (3) mainly depends on the effective arm length r, with a minor contribution coming from the deflection of the spring mechanism which is bounded to be in the range ±0.2 rad. The lever arm is adjusted through a ball screw guiding mechanism driven by the motor M2 such that r = r0 − nθ2
I q¨ + N q˙ + τE = τext B1 θ¨1 + φ1 θ˙1 − τE = µ1 u1 B2 θ¨2 + φ2 θ˙2 + τr = µ2 u2
(5)
Symbol I N B1 D1 ν1 Kt1 Ke1 Rm1 B2 D2 ν2 Kt2 Ke2 Rm2 ks n r0
Value 0.1 0.15 0.0575 0.001 50 0.11 0.11 3.89 6.8294e-5 1.3e-5 23 0.0136 0.0136 13.8 80000 2.5/2π 0.1
Unit kgm2 N ms/rad kgm2 N ms/rad N m/A V s/rad Ω kgm2 N ms/rad N m/A V s/rad Ω N/m mm/rad m
III. P ROBLEM S TATEMENT The nonlinear plant in (5) can be written as
(4)
where θ2 is the angular position for the motor (M2 ), r0 is the initial length of the lever arm and n is the transmission ratio between the motor and the ballscrew. The schematic of AwAS is presented in Fig. 2 and the dynamics, neglecting the gravity contribution, are described as
Schematic of the AwAS.
TABLE I PARAMETER FOR THE AWAS
(1)
τE = ks r2 sin(2θs )
τ ext
N
x˙ = A(¯ σ (t))x + Bu + Bext τext + ψ(θs , θ2 )
(7)
σ ¯ (t) = 2ks (ro − nθ2 (t))2
(8)
with where A ∈ R6×6 , B ∈ R6×2 , Bext ∈ R6×1 , u ∈ R2×1 denotes the control input, τext the external torque, the state is x = [q q˙ θ1 θ˙1 θ2 θ˙2 ]T . (9) and ψ(θs , θ2 ) ∈ R6×1 (see Appendix). The function ψ(θs , θ2 ) is bounded, since θ2 ∈ [0, θ2max ] and |θs | ≤ 0.2.
2786
Property 1. ψ(θs , θ2 ) is continuously differentiable in any open connected set containing the origin, ψ(0, 0) = 0, ∂ψ(0, 0)/∂q = 0, ∂ψ(0, 0)/∂θ1 = 0 and ∂ψ(0, 0)/∂θ2 = 0. This implies that for any given ξ > 0, there is a χ > 0 such that ||ψ(q − θ1 , θ2 ) − ψ(ˆ q − θˆ1 , θˆ2 )|| ≤ ξ||x − x ˆ||
(10)
for all ||x|| < χ, ||ˆ x|| < χ. Equation (10) can also be expressed as lim||x||→0
||ψ(q − θ1 , θ2 )|| =0 ||x||
(11)
In addition, each non-zero entry in the function ψ(θs , θ2 ), can be written such as ψ(θs , θ2 ) = f (θs )g(θ2 ) where g(θ2 ) is bounded and f (θs ) satisfy the Property 1. Following this, the nonlinear system in (7) is reformulated as a parameterized linear system x˙ = A(¯ σ (t))x + Bu + Bext τext
(12)
and further decomposed in two separate systems x˙ 1 = A1 (¯ σ (t))x1 + Bu1 u1 + Bext1 τext1
(13)
x˙ 2 = A2 x2 + Bu2 u2
(14)
and where x1 = [q q˙ θ1 θ˙1 ]T and x2 = [θ2 θ˙2 ]T , while u1 and u2 are the control input signals associated with the motors M1 and M2 respectively. The resulting linear parameter varying system (LPV) in (12) depends on σ ¯ (t) which satisfies the following assumptions: Assumption 1. σ ¯ (t) is rate bounded ˙ (t)| < ρ¯ |¯ σ
(15)
(16)
The problem that is considered in the following section is therefore to control the system in (7) in order to guarantee a reasonable tracking of the link position q while damping the oscillations related to the flexible transmission. To simplify the notation, the explicit dependence on time t will often be omitted, so that σ ¯ (t) is written as σ ¯. IV. P OSITION C ONTROL The controller synthesis problem for the nonlinear systems (7) is divided into a set of local linear sub-problems. For this purpose, the LPV system in (13) is initially considered, by taking advantage of the decoupled dynamics (13,14) resulting from the parameterized linearization of the original system. Then a set of linear time invariant systems (LTI) is formulated such as x˙ 1 = A1 (¯ σ i )x1 + Bu1 u1
(17)
by evaluating (13) along a set of fixed values of σ ¯i ∈ [¯ σm , σ ¯M ] while setting the external disturbance τext1 equal to zero. A state feedback control u1 is therefore obtained u1 = G(¯ σ i )qd − Klqr (¯ σ i )x1
0
subject to the dynamics (17), with Q(¯ σ i ) > 0 and R(¯ σi ) > 0 symmetric weighting matrices. The resulting closed loop system results to be x˙ 1,cl = A1cl (¯ σ i ) x1,cl + Bu1 G(¯ σ i )qd
(20)
A1,cl (¯ σ i ) = A1 (¯ σ i ) − Bu1 Klqr (¯ σ i ). Following this, given the linear controller family in (18), the objective is to design a continuous controller signal for the LPV system in (13). For this purpose a gain scheduling is designed through a polynomial fitting of both the controller gains Klqr (¯ σ i ) and the feedforward terms G(¯ σ i ) as a function of σ ¯ ∈ [¯ σm , σ ¯M ] such that Pm ˆ lqr (¯ K σ ) = h=0 (¯ σ ) h βh P (21) m ˆ G(¯ σ ) = h=0 (¯ σ )h γh
ˆ lqr (¯ ˆ σ ) are the resulting gain and feedwhere K σ ) and G(¯ forward term, respectively (Fig. 3(a)). Given (21), the gain scheduled closed loop system is formulated as ˆ σ )qd x˙ 1,cl = Aˆ1cl (¯ σ ) x1,cl + Bu1 G(¯
(22)
ˆ lqr (¯ σ ). Aˆ1,cl (¯ σ ) = A1 (¯ σ ) − Bu1 K
Assumption 2. σ ¯ (t) takes values in a bounded interval σ ¯ (t) ∈ [¯ σm , σ ¯M ]
where qd is the link desired position, G(¯ σ i ) is a feedforward term included in the control to compensate for the steady state error and Klqr (¯ σ i ) = [Kq Kq˙ Kθ1 Kθ˙1 ] is the feedback gain vector. This is obtained through the solution of the Riccati equation to minimize a set of performance indexes Z ∞ J(¯ σi ) = (xT1 Q(¯ σ i )x1 + uT1 R(¯ σ i )u1 )dt (19)
(18)
To verify the goodness of the fitting (21), comparison in link step response between the closed loop performance of (22) and (20) is conducted. The difference in link positioning is showed in Fig. 3(b). Based on the results, the fourth order polynomial function was considered a reasonable approximation of the LQR gains (Klqr (¯ σ i )), without further increase in the computational complexity. In addition, comparison of the gain scheduling performance when applied to the nonlinear system and the linear system is showed in Fig. 3(c). Also in this case, the difference in link positioning is not significant. On the other hand, degradation in closed loop performance can arise if the true value of σ ¯ (t) becomes substantially different form the values considered in the design of the LQR controller. As a consequence, an accurate estimate of the stiffness is recommended (within 30 % of the actual value). a) Performance Index: The aim of the LQR control is to find the control law which minimizes the time integral of the perturbation energy, while keeping the time integral of the control effort as low as possible (19). Any selection of the weighting matrices Q(¯ σ i ) and R(¯ σ i ) diagonal and positive definite is usually acceptable. However, the large variation in the plant dynamics requires the matrix elements to reflect the necessary scaling to weight the variables in the performance index so to achieve a good link tracking control while keeping low undesirable vibrations.
2787
−3
−4
x 10
Kq Kq˙ Kθ1 Kθ˙1
5
−0.5
4
−1
3
−1.5
2
−1500
[rad]
−1000
[rad]
K lqr [Volt/rad]
−500
x 10
0
0
−2
1
−2.5
0
−3
−1
−2000
−2500
−3.5
−2
−4
−3000 0
200
400
600
800
1000
1200
1400
−3 0
1600
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.5
1
¯ [Nm/rad] K
time [s]
time [s]
(a)
(b)
(c)
1.5
σ¯ [Nm/rad] σ [Nm/rad] 200
200
1600 1460
150
110
1460
1460
100
1280
980 820
0
660
−50
500
−100
Imaginary [rad/s]
1140 50
1300 100 1140 50
980 820
0
660
−50
180 20 −160
−140
−120
−100
−80
−60
Real [1/s]
(d)
−40
−20
0
90
1100
80 920 70 740 60 560 50
500
380
40
−100
340 −150
Magnitude [db]
1300
Imaginary [rad/s]
1600
1600
150
100
−200 −180
120
σ [Nm/rad]
340
200
30
−150
180
−200
20 −250
−200
−150
−100
−50
0
20 −2 10
20 10
−1
10
0
10
1
10
2
10
3
ω [rad/s]
Real [1/s]
(e)
(f)
ˆ lqr (¯ Fig. 3. AwAS controller design. (a) Control gains K σ (t)) obtained through polynomial fitting. (b) Difference in link positioning between the closed loop system in (22) and (20) for σ ¯ i = 200 Nm/rad (- -) and σ ¯ i = 1600 Nm/rad (-). (c) Difference in link positioning when the gain scheduling is applied to the nonlinear and linear system for σ ¯ i = 200 Nm/rad (- -) and σ ¯ i = 1600 Nm/rad (-). Eigenvalues location for the open loop (d) and closed loop (e) system, calculated for increasing values of the stiffness σ ¯ i . (f) Frequency analysis of the stiffness perceived at the link (S) formulated as in (25) for increasing values of the stiffness σ ¯ i when the system is controlled with gain scheduling.
For this purpose, the analysis of the eigenvalues location of (17) calculated for several values of σ ¯ i is conducted. When σ ¯ i = 20 Nm/rad the system has four eigenvalues λ1 = −177.1, λ2 = −1.2 + 7.8i, λ3 = −1.2 − 7.8i and λ4 = 0 in the open left-half plane. As the stiffness increases, the location of the eigenvalues changes as it is showed in Fig. 3(d). In detail, apart from the eigenvalue placed in zero which is not affected by the plant variations, the negative real eigenvalue move towards lower frequencies while the displacement of the complex eigenvalues show a decreasing damping of the system as the stiffness increases. Based on this, the matrices elements for (19) are selected such as σi β γ σ ¯i δ Q(¯ σ i ) = diag α¯ , R(¯ σi ) = 1 (23) with α, β, γ and δ being constants. Note that, when γ and δ are significantly higher than α, β, which in turns implies that the penalties at the motor side are higher than those at the link, this produces significant oscillations of the system since the eigenvalues with complex part become slower than those with negative real part. Alternatively, when increasing the penalties at the link side with respect to those at the motor, the performance of the system in terms of bandwidth reduces. For the AwAS application, the constant values are ultimately assigned such as α = 15097, β = 3312, γ = 302 and δ = 57, in this way the closed loop behavior, which is showed in (Fig. 3(e)), has increasing performance for increasing values of the σ ¯ i with an almost constant damping ratio of 0.6 in the range
of stiffness variation. In addition, it can achieve a theoretical bandwidth in the link positioning for small amplitude input signal ranging from 2 Hz (¯ σ i = 20 Nm/rad) to 13 Hz (¯ σi = 1600 Nm/rad), on the basis of the stiffness value. V. S TIFFNESS CONTROL The strategy adopted for controlling the stiffness perceived at the output link was initially focused in expressing this stiffness in terms of reference values for the elastic transmission, taking into account the effects of the positioning gains coming from the controller. In detail, the stiffness perceived at the output link (S) depends on how the virtual (controller) and real springs (elastic transmission) connect respectively. This is clarified when considering the link stiffness S=
δτext δq
(24)
as the variation of the external torque over the link displacement, which for the system in (22) becomes ˆ q + sµ1 σ ˆ q˙ −¯ σ 2 + µ1 σ ¯K ¯K , ˆθ ˆ˙ )+σ ¯ + µ1 K B1 s2 + s(φ1 + µ1 K 1 θ1 (25) where it is straightforward to note the relationship among the S, σ ¯ and the LQR gains. The relationship in (25) can be further simplified when considering that for a given value of mechanical stiffness σ ¯ , the link stiffness variation S is S = Is2 + N s + σ ¯+
2788
_
not significant in the low frequency region (Fig. 3(f)). As a consequence, (25) can be reformulated such as λ=1+
PID
_ σ_ qd _ _ _ _ _ _ u1 . _ G(σ)q - K (σ)x x =A (σ)x +B
(26)
1
d
lqr
1
1
1
1
u1
u1
_
. . x=[q q θ1 θ1] T 1 _
_
which is a reasonable approximation in the range of frequencies up to 10 rad/s (Fig. 3(f)). Note that (26) can be considered as the series connection of two springs λ¯ σ, ˆ θ and that if λµ1 K ˆ θ >> λ¯ λµ1 K σ then S ≈ λ¯ σ . This 1 1 in turn implies that if the magnitude of the controller gain ˆ θ is considerably large, then σ ¯ is the main contributor K 1 to the final link stiffness value. This is also a reasonable result, since the controller should mainly take advantage of the flexible element in the transmission to achieve the desired behavior. The interpolation stage of the gain scheduling is then exploited to obtain the reference value for the mechanˆ q ) are ˆθ , K ical stiffness σ ¯d . Since the controller gains (K 1 polynomial functions of σ ¯ as in (21), it is possible to express the desired link stiffness value Sd in (26) as a function of the only mechanical stiffness σ ¯d . As a consequence, it is possible to derive the mechanical stiffness that combined with the positioning gain determines the desired stiffness behavior at the output link. Note that, the polynomial function is limited up to the fourth order also to reduce the computational complexity in solving (26). This indeed presents multiple solutions, but only one among them has a physical meaning when the link stiffness Sd ∈ [32, 1877] Nm/rad. b) Mechanical stiffness regulation: The controller for the mechanical stiffness adjustment is developed to quickly achieve the desired stiffness value at the elastic transmission. Given σ ¯d from (26), the corresponding desired position θ2d at the motor M2 is obtained from (4) and (8) such as r σ ¯d 1 r0 − . (27) θ2d = n 2ks
S(s=0)
_
,
ˆq K ˆθ K
σd
_
S(s→0) =
ˆ σ ¯µ K λ σ¯ +µ1 Kˆθ1 1 θ1
Sd
Fig. 4.
Schematic of the control strategy.
control strategy when the scheduling variable is varying [14]. For this case, it is sufficient to ensure that the variation of the scheduling variable is slow [14]. This essentially requires to find the largest possible bound ρ¯, introduced in the Assumption 1, for which the stability of the closed loop in (22) is guaranteed. Based on the approach described in [19], the largest value of ρ¯ able to guarantee the stability of (22) depends on the minimum stiffness σ ¯m value, and for this application results to be ρ¯ = 308 [Nm/rad s] (for σ ¯m = 20 [Nm/rad]). Note that the stability holds also for the cases when the mechanical stiffness variation achieves higher values with respect to ρ¯. The bounded rate is indeed a sufficient condition. The stability can be extended to the total system comprehensive of (14) when controlled with a stable PID controller. However, this does not necessarily imply that the closed loop formulation of the nonlinear system in (7) is stable. To prove the stability of (7), it is necessary to recall that the nonlinearities ψ(θs , θ2 ) are sufficiently smooth and small in the neighborhood of the operating point, as it is discussed in the Property 1. Based on this, it is possible to prove that the closed loop nonlinear system is locally stable [20].
and then sent as input signal to a standard PID position control scheme. This is implemented on the system described in (14) for positioning the ballscrew and as a consequence the springs, with respect to the center of rotation of the joint. Note that the mechanical stiffness setting is not instantaneous and it mostly depends on the time it takes for positioning the ballscrew in the desired position with respect to the initial one. In the details, the motor M2 takes about 5 s to decrease the stiffness from the value of 1600 Nm/rad to 20 Nm/rad and the bandwidth of the closed loop system to small input signals is 23.3 rad/s. The overall control scheme is presented in Fig. 4. Fig. 5.
VI. S TABILITY The proof of stability of the overall control strategy is particularly laborious and only an outline is provided in this work due to limited space. Given the stability of the gain scheduling controller when the system is operating in the vicinity of a designed operating condition [19]. This in turn implies that having a set of stable LQRs as in (20), verified through the approach described in [17], the gain scheduling is stable in the neighborhood of these points. However, this does not directly extend to the gain scheduling
Experimental Setup.
VII. E XPERIMENTAL R ESULTS The AwAS experimental platform is used for verifying the effectiveness of the control strategy (Fig. 5). The platform is provided with two incremental encoders (40000 counts/rev) placed between and after the elastic transmission and an incremental encoder on the motor M2 side (before gear) with resolution of 512 (counts/rev). The prototype also includes a torque sensor. The control strategy is implemented on a
2789
custom DSP-board running at 1 kHz. During the experiment the platform is not subjected to gravity and a weight of 2 kg is attached at the link.
0.6 0.5 0.4
[rad]
0.3 0.2
Error[rad]
0.015 0.01
0.1
0.005
0
0
−0.1
θ1 q qdes 99
100
101
102
103
104
105
106
time [s]
−0.005
(a)
−0.01 −0.015 20
−0.02 15
0
0.5
1
1.5
2
2.5
3 10
Torque [Nm]
time [s]
Fig. 6. Error in link sine wave tracking (qdes − q) executed at 1 Hz when the stiffness at the transmission is constant with values σ ¯ = 200 Nm/rad (-) and σ ¯ = 1600 Nm/rad (-).
5 0 −5 −10 −15 −20 98
0.025
100
101
102
time [s]
103
104
105
106
(b)
0.02 0.015
Fig. 8. Link step response performance when an external perturbation is applied (a). Torque measurements (b).
0.01
[rad]
99
0.005 0 −0.005 −0.01 −0.015 −0.02 63.5
64
64.5
time [s]
65
65.5
66
65
65.5
66
(a) 1300 1250
σ ¯ [Nm/rad]
1200 1150 1100 1050 1000 950 63.5
64
64.5
time [s]
(b) Fig. 7. Error in link sine wave tracking executed at 1 Hz (a), when the stiffness σ ¯ is changing at the frequency of 0.1 Hz (b).
Initially, tests in tracking a sine wave reference for the link position (qd ) when the stiffness (¯ σ ) is kept constant are conducted. Note that the stiffness is computed on the basis of (8), however there is not a significant difference with the one obtained with (3), given the bounded spring deflection. The tracking error (qd − q) when the reference has the amplitude of 0.22 rad and the frequency of 1 Hz is showed in Fig. 6. Clearly, the closed loop system is able to smoothly follow the input reference value with a limited error even for significant different values of the stiffness at the mechanism. The error amplitude depends on the bandwidth of the controlled system which, on the other hand, changes on the basis of the stiffness value at the elastic transmission, as expected from
the simulation results. Then, position tracking experiments are conducted when the stiffness is continuously varying. Figures 7(a) and 7(b) show the tracking error and stiffness, respectively. Note that the stiffness sine wave has amplitude of 460 Nm/rad and frequency of 0.1 Hz to be within the bounds of the maximum stiffness variation (¯ ρ) defined by the stability criteria. The results showed the tracking ability of the controlled system in spite of the significant changes in the plant dynamics, also confirmed by the substantial variations in the positioning gains associated with the given reference stiffness (Fig. 3(a)). Finally, the ability of controlling the output link stiffness through combination of both the stiffness at the transmission ˆ lqr (¯ (¯ σ ) and the controller gains (K σ )) is tested. In the experiment, the desired link stiffness is set at Sd = 70 Nm/rad and the mechanical stiffness is derived through (26) such as σ ¯d = 46.9 Nm/rad. The stiffness (S) was verified through position and torque measurements, by manually hitting and holding the link for few seconds once it had reached its step reference position. The result is presented in Fig. 8(a), by taking an average of the link position data during the time interval [102.35, 104.62] s, also considering the corresponding torque measurements in Fig. 8(b) the stiffness obtained at the link is S = 71.35 Nm/rad. VIII. C ONCLUSIONS This paper discussed an LQR-based gain scheduling approach for the link position and stiffness control of actuators with varying stiffness. In this work the actuator’s nonlinear dynamics is initially expressed with a LPV formulation. Then a set of LQRs was designed for a set of LTI systems derived from the original LPV system. Finally, a gain scheduling
2790
strategy was formulated through polynomial interpolation of the LQR gains. Comparisons in simulation demonstrated the goodness of the gain fitting in guaranteing the system behavior specified through the LQR design. In addition, further analysis showed the capability of the gain scheduling to cope with the nonlinearities of the original actuation’s plant. The link positioning was obtained through adjustment of the controller gains depending on the stiffness values set at the transmission. The control of the link stiffness was obtained through the combined action of a PID controlled stiffness adjustment at the transmission and the gain scheduling. For this purpose, the formulation of the stiffness at the output link, showing a clear connection of the positioning control gains with the stiffness at the elastic transmission, was exploited. The overall control strategy was verified first by simulations, then by experiments with the Actuator with Adjustable Stiffness (AwAS). The results confirmed the ability of the strategy to cope with the variability of the plant showing remarkable performance in tracking, while actively damping the vibrations. Experiments also verified the effectiveness of the strategy in controlling the output link stiffness in the case of slow external perturbations. However it is important to mention that the performance of the output link stiffness controller is ultimately limited by the time the mechanism takes for setting the required stiffness. Future work will consider the extension of this control strategy to multi-dof robotic platforms using a decentralized control approach. IX. ACKNOWLEDGMENTS The authors gratefully acknowledge the precious help of Phil Hudson and Giuseppe Sofia. This work is supported by the European Community, within the FP7 ICT-287513 SAPHARI project. A PPENDIX When considering the dynamics of the AwAS in (5), by adding and subtracting the term 2ks r2 (q − θ1 ) to the first two equations it is possible to reformulate (5) such as x˙ = A(¯ σ )x + Bu + Bext τext + ψ(θs , θ2 )
(28)
where
A=
B=
A1
z
0 − σI¯ 0 σ ¯ B1
1 − NI 0 0
}|
0
0 0 1 φ1 −B 1
σ ¯ I
0 − Bσ¯1 02×4
0 0 Bu1 0 µ1 02×1
Bu2
04×2 (29) A2 z }| { 0 1 φ2 0 −B 2
04×1
B1
{
0 µ2 B2
Bext
0
1 I 0 = 0 0 0
and
ψ(θs , θ2 ) =
0 2
2
2ks r θs −ks r sin(2θs ) I
0
−2ks r 2 θs +ks r 2 sin(2θs ) B1
0
2ks nr[sin(θs )]2 B2
(30)
with x = [q q˙ θ1 θ˙1 θ2 θ˙2 ]T , µi = −νi Kti /Rmi , i ∈ [1, 2], θs = q − θ1 , σ ¯ = 2ks r2 and r = (r0 − nθ2 ). R EFERENCES [1] A. Bicchi and N. Tonietti. Fast and soft arm tactics: Dealing with the safety- performance tradeoff in robots arm design and control. Rob. and Aut. Mag., pages 22–33, 2004. [2] J. W. Hurst, J. E. Chestnutt, and A. A. Rizzi. An actuator with physically variable stiffness for highly dynamic legged locomotion. Int. Conf. on Rob. and Aut., 2004. [3] S.A. Migliore, E.A. Brown, and S.P. DeWeerth. Novel nonlinear elastic actuators for passively controlling robotic joint compliance. ASME Journ. Mech. Design, pages 406–412, 2007. [4] R. Schiavi, G. Grioli, S. Sen, and A. Bicchi. Vsaii: A novel prototype of variable stiffness actuator for safe and performing robots interacting with humans. Int. Conf. on Rob. and Aut., pages 2171–2176, 2008. [5] M. G. Catalano, G. Grioli, F. Bonomo, R. Schiavi, and B. Bicchi. Vsahd:from the enumeration analysis to the prototypical implementation. Int. Conf. on Rob. and Aut., pages 3285–3291, 2010. [6] S. Wolf and G. Hirzinger. A new variable stiffness design: matching requirements of the next robot generation. Int. Conf. on Rob. and Aut., pages 1741–1746, 2008. [7] B. S. Kim and J. B. Song. Hybrid dual actuator unit: A design of a variable stiffness actuator based on an adjustable moment arm mechanism. Int. Conf. on Rob. and Aut., pages 1655–1660, 2010. [8] B. Vanderborght, N. Tsagarakis, C. Semini, R. Van Ham, and D.G. Caldwell. Maccepa 2.0: Adjustable compliant actuator with stiffening characteristic for energy efficient hopping. Int. Conf. on Rob. and Aut., 2009. [9] A. Jafari, N.G. Tsagarakis, and D. G. Caldwell. Awas-ii: A new actuator with adjustable stiffness based on the novel principle of adaptable pivot point and variable lever ratio. Int. Conf. on Rob. and Aut., pages 3561–3569, 2011. [10] N.G. Tsagarakis, I. Sardellitti, and D. G. Caldwell. A new variable stiffness actuator (compact-vsa): Design and modelling. Int. Conf. on Int. Rob. and Sys., pages 378 – 383, 2011. [11] G. Tonietti, R. Schiavi, and A. Bicchi. Design and control of a variable stiffness actuator for safe and fast physical human/robot interaction. Int. Conf. on Rob. and Aut., pages 526–531, 2005. [12] G. Palli, C. Melchiorri, and A. De Luca. On the feedback linearization of robots with variable joint stiffness. Int. Conf. on Rob. and Aut., pages 1753–1759, 2008. [13] A. De Luca, F. Flacco, A. Bicchi, and R. Schiavi. Nonlinear decoupled motion-stiffness control and. collision detection/reaction for the vsaii variable stiffness device. Int. Conf. on Int. Rob. and Sys., pages 5487–5494, 2009. [14] J. S. Shamma and M. Athans. Gain scheduling: Potential hazards and possible remedies. IEEE Control Systems Magazine, pages 101–107, 1992. [15] A. Albu-Schaffer, S. Wolf, O. Eiberger, Haddadin S., Petit F., and Chalon M. Dynamic modelling and control of variable stiffness actuators. Int. Conf. on Rob. and Aut., pages 2155–2162, 2010. [16] G.F. Franklin, J. D. Powell, and A. Emami-Naeini. Feedback control of dynamic systems. Prentice Hall. [17] R.E. Kalman. Contribution to the theory of optimal control. Int. Conf. on Rob. and Aut., pages 102–119, 1960. [18] A. Jafari, N. Tsagarakis, B. Vanderborght, and D. G. Caldwell. A novel actuator with adjustable stiffness (awas). Int. Conf. on Int. Rob. and Sys., pages 2782–2788, 2010. [19] W. J. Rugh and J. S. Shamma. Research on gain scheduling. Automatica, pages 1401–1425, 2000. [20] H. K. Khalil. Nonlinear systems. Prentice Hall, New Jersey, 2002.
2791
