Proxy based position control for flexible joint robot with link side energy feedback

Robotics and Autonomous Systems 121 (2019) 103272

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Proxy based position control for flexible joint robot with link side energy feedback ∗

Lei Sun , Wen Zhao, Wei Yin, Ning Sun, Jingtai Liu Institute of Robotics and Automatic Information System of Nankai University, No. 38 Tongyan Road, Jinnan District, Tianjin, China Tianjin Key Laboratory of Intelligent Robotics, No. 38 Tongyan Road, Jinnan District, Tianjin, China

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Article history: Received 28 February 2019 Received in revised form 31 August 2019 Accepted 2 September 2019 Available online 5 September 2019 Keywords: Flexible joint robot Position control Energy feedback Proxy-based sliding mode control

a b s t r a c t In this paper, a set-point regulation scheme for flexible joint robot (FJR) with link side energy (LSE) feedback is introduced. The drawbacks of traditional PD-type control laws are analyzed first. On this basis, a nonlinear regulator is designed by combining proxy based sliding mode control with LSE feedback, which is targeted at enhancement of vibration suppression. Asymptotic stability of the closed-loop system as well as boundedness of each signal are guaranteed with Lyapunov analysis. Furthermore, vibration suppression ability and robustness against external disturbances of the proposed method is validated on a self-built FJR platform. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Flexible joint robot is a more generic model for traditional industrial robot [1], since it takes flexible transmissions, like harmonic drive, belt-pulley, joint torque sensors into consideration [2]. However, most robots have been designed to be as stiff as possible [3] due to the complexity of controlling flexible elements [4], and one of the unwanted response is the oscillatory behavior at end-effector while there is no specific control action [5]. Therefore, if high performance is preferred, flexible joint, instead of rigid joint mathematical model should be employed in controller design [6]. On the contrary, recently, flexible actuation, like series elastic actuator [7], variable impedance/stiffness actuator [8] has been deliberately adapted in robot design, intended for human– robot interaction, like Baxter [9], DLR Hand Arm System [10], the CompActTM arm of Italian Institute of Technology (IIT) [11] and NASA’s ‘‘Valkyrie’’ [12]. The reasons are manifold, e.g., lowering the contact stiffness among robot, human and environment, thus attenuating potential injury [13]; inertia decoupling between actuator and driven link, and further reducing kinetic energy involved in undesired collisions [14]; shock tolerance [15], etc. As a result, FJR has been regarded as one of the promising equipments for next generation industrial applications [16]. However, the flexible elements, whether from traditional transmissions, or safety-oriented mechanical design, will bring ∗ Corresponding author at: Institute of Robotics and Automatic Information System of Nankai University, No. 38 Tongyan Road, Jinnan District, Tianjin, China. E-mail address: [email protected] (L. Sun). https://doi.org/10.1016/j.robot.2019.103272 0921-8890/© 2019 Elsevier B.V. All rights reserved.

much more difficulties to position control comparing with the rigid ones [17]. The related model is decoupled into motor side dynamics and link side dynamics by flexible elements [18], and both of them are second order without considering electrical characteristic [19]. Therefore, the order of the FJR model is twice that of the rigid joint robot model [20]. This feature also results in the under-actuated performance [21,22], since the number of dimensions of the configuration space is twice the number of dimensions of the control input space [23]. Moreover, the oscillatory behavior is easy to be inspired which is not desired during motion [24], and it may further degrade performance and even damage the robots [25]. Besides, the nonlinearity, the coupling between motor side and link side [18] as well as the non-collocated feature [26] also impose more difficulties to the position control of FJR [27]. The position control for FJR has attracted many scholars’ attention due to the increased complexity and demands for applications [28]. The first decade (from late 1980s to late 1990s) witnessed theoretical fruit in the control of FJR [29], like singular perturbation [30], feedback linearization [31], backstepping [32] etc. In 1991, Tomei proposed a simple PD controller along with constant gravity compensation (CGC) [6]. Since then, different types of PD regulators have been presented due to its simplicity and practicability [33]. Based on Tomei’s method, De Luca et al. replaced the CGC with online gravity compensation (OGC) [34] and experiment results on Dexter robot illustrated the improvements. However, both CGC and OGC require that the proportional gain should be large enough to guarantee closed-loop stability. Thus, Albu-Schäffer et al. presented a PD-type controller that is augmented by non-collocated feedback to shape the kinetic

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energy, and the requirement is merely that proportional gain should be positive [35]. To accurately match the behavior of the links as if they were moving in the absence of gravity in any dynamic condition, a PD-type regulator with exact gravity cancellation (EGC) was proposed [36]. However, it requires exact model knowledge and full state feedback due to the second order derivation of gravity. Whether gravity compensation or cancellation, the basic assumption is that gravity is exactly known. If this condition is not satisfied, PD control equipped with a classical integral action on link position error became effective [37]. Based on these PD-type regulators, some more approaches have been designed to achieve better performance. In 2016, a proxy-based controller was presented to enhance the safety-related performance of FJR [38]. In the same year, an interconnection and damping assignment passivity-based approach was introduced to achieve precise control [39]. Based on the state-dependent Riccati equation (SDRE), a position and vibration control method was proposed in [40] and Magnetorheological dampers was applied to improve the control performance. In the progress of increasing positioning accuracy, friction is an obstacle that should be overcome. The corresponding methods include adaptive friction compensation [41], friction/disturbance observer(DOB) [42,43], model-based friction compensation [44,45], extended state observer [46,47], sliding mode like control with DOB [48], etc. Some other methods, like prescribed performance control [49], resonance ratio control [50], adaptive command-filtered backstepping control [51] and neural network based control [52] are also used to realize position control of FJR, and some achievements have been made. All of the above literatures can stabilize closed-loop system, but there are still some certain issues require attention. For instance, some methods are highly model-dependent [36,53]; A few others are only suitable for single degree of freedom [46,50,54]. Most of all, literatures above, including PD-type controllers, lack effective scheme for suppressing residual vibration [55]. In response to the limitations above, we present a proxybased position control method for FJR with LSE feedback. Most PD-type regulators only use motor side variables, while link side can only experience damping oscillation until steady state, and the damping feature is only caused by friction. The oscillation behavior of link side indicates that LSE, including kinetic energy, gravitational potential energy and elastic potential energy is not zero. Therefore, with LSE feedback, as long as it is not zero, the control input will get a certain value which is helpful to attenuate vibration. Besides, another additional term is introduced into control law which will also help to stabilize the closed-loop system. By combining LSE feedback, the accuracy and residual vibration suppression of the safe-oriented proxy-based position control are further improved. The rest of this paper is organized as follows. In Section 2, problem formulation, including general dynamics, some properties and reasonable assumptions as well as control objective is briefly explained. Controller design and stability analysis procedures are provided in Sections 3 and 4, respectively. In Section 5, some comparing experiments are implemented to verify effectiveness of the proposed control scheme. The paper ends with conclusion given in Section 6.

by joint elasticity; for linear model, τc (φ ) = K φ with K = diag(K1 , K2 , . . . , Kn ) representing stiffness matrix; M(q), J ∈ Rn×n denote link side and motor side inertia matrix, respectively, and J = diag{J1 , J2 , . . . , Jn }; C (q, q˙ )q˙ ∈ Rn is Coriolis and centripetal torque; g(q) ∈ Rn denotes gravity torque; τfl (q, q˙ ), τfm (θ , θ˙ ) ∈ Rn are link side and motor side friction; τm ∈ Rn is the control input. In order to facilitate the controller design and stability analysis, the following properties and assumptions are used, which can be found in most robot-related literatures. P1: Inertia matrix M(q) is symmetric and positive definite, that is,

∀ξ , q ∈ Rn : M(q) = M T (q), ξ T M(q)ξ ≥ 0

(2)

− C (q, q˙ ) is skew-symmetric and ] [ 1 ˙ M(q) − C (q, q˙ ) ξ = 0 ∀ξ , q, q˙ ∈ Rn : ξ T

(3)

P2: The matrix

˙

1 M(q) 2

2

Let Ug (q) and Uc (φ ) be potential energy functions related to gravity and elastic deformation, respectively, thus,

∂ Ug (q) ∂ Uc (φ ) , τ c (φ ) = (4) ∂q ∂φ Hessians of Ug (q) and Uc (φ ), i.e., the Jacobians of g(q) and τc (φ ), are defined as g(q) =

Jg (q) =

∂ 2 Ug (q) ∂ g(q) ∂ 2 U c (φ ) ∂τc (φ ) = , Jc (φ ) = = 2 ∂q ∂q ∂φ 2 ∂φ

P3: Both g(q) and Ug (q) are formed by trigonometric functions of variable q; thus, the following inequalities hold:

∀ q ∈ Rn , ∃ ζ1 > 0 : |Ug (q)| < ζ1 ∀ q ∈ Rn , ∃ ζ2 > 0 : ∥Jg (q)∥ < ζ2

(6)

The joint torque generated by elastic deformation should be large enough for two purposes: one is to maintain the link postures under the gravity field, and the other is to drive the link side to the desired position during transient state. Statement above can be expressed as the following assumption: A1: The minimum eigenvalue of Jc (φ ) should be larger than the Jacobian Jg (q) of the gravity torque g(q), that is,

∀ξ , φ ∈ Rn : ξ T Jc (φ )ξ > ζ2 ∥ξ ∥2

(7)

In the linear FJR system, Jc (φ ) equals the stiffness matrix K . As a basic requirement for the joint stiffness design, Assumption A1 is physically reasonable and has been adopted in a large number of previous works [6,34,35]. Let Ut (q, θ ) = Ug (q) + Uc (φ ) be total potential energy of the FJR system. Based on property P3 and assumption A1, the following property can be obtained: P4: The Hessian of the total potential energy function Jt (q, θ ) = ∂ 2 Ut (q,θ ) is positive definite, i.e., ∂ q2

∀q, θ, ξ ∈ Rn : ξ T Jt (q, θ )ξ > 0

(8)

In steady state, the torque generated by elastic deformation is equal to gravity, that is

2. Problem formulation

∀θ ∈ Rn , ∃ q¯ ∈ Rn : g(q¯ ) = τc (θ − q¯ )

Based on Spong’s assumptions [18], the following simplified FJR model has been accepted widely [6,34,35,56]:

In the perspective of energy, (9) is equivalent to

M(q)q¨ + C (q, q˙ )q˙ + g(q) + τfl (q, q˙ ) = τc (φ ) J θ¨ + τfm (θ , θ˙ ) + τc (φ ) = τm

(1)

where q, θ ∈ R are link side and motor side position, respectively; φ = θ − q, and τc (φ ) denotes torque generated n

(5)

∀θ ∈ Rn , ∃ q¯ ∈ Rn :

(9)

∂ Ut (q, θ ) ⏐⏐ =0 ⏐ q=¯q ∂q

(10)

From (10), q¯ can be calculated via Newton–Raphson method, as follows: q¯ j = q¯ j−1 − Jt−1 (q¯ j−1 , θ ) g(q¯ j−1 ) − τc (q¯ j−1 , θ )

[

]

(11)

L. Sun, W. Zhao, W. Yin et al. / Robotics and Autonomous Systems 121 (2019) 103272

where j denotes iteration index. When ∥¯qj − q¯ j−1 ∥ < e, iteration stops where e is a tolerant error. Combining property P4 with (10), the following property can be obtained: P5: Ut (q, θ ) gets the global minimum value at point (q¯ , θ )

∀q, θ ∈ Rn : Ut (q¯ , θ ) ≤ Ut (q, θ ) From (10), it is clear that

∂ q¯ ∂θ

(12)

= Jt (q¯ , θ )Jc (θ − q¯ ). Thus, −1

q˙¯ = Jq¯ (q¯ , θ )θ˙

As a result, dynamics of proxy error e˙ p can be derived as e˙ p = Kd−1 Γ sat[Γ −1 (Kd S −1 ψ + Kp ep )] − Kd−1 Kp ep

Jq¯ (q¯ , θ ) = Jt (q¯ , θ )Jc (θ − q¯ ) −1

(14)

A2: As assumed in [6], frictions of both link and motor side are dissipative, that is,

∀ q, q˙ ∈ R : q˙ τfl (q, q˙ ) ≥ 0 ∀ θ , θ˙ ∈ Rn : θ˙ T τfm (θ, θ˙ ) ≥ 0 n

T

(15)

if and only if q˙ = θ˙ = 0, the equality holds. Regulation task of this paper is designing a control input τm for (1) based on the properties and assumptions above, so that link position q can be stabilized to a desired constant value qd , with residual vibration being effectively suppressed, while all closed-loop signals remain bounded. 3. Controller design The following functions will be used throughout this paper: z /|z | [−1, 1]

{ sgn(z) ≜ sat(z) ≜

if z ̸ = 0 if z = 0

z max(1, |z |)

eq¯ = qd − q¯ , ψ = eq¯ − ep + S e˙ q¯

(18)

(19)

(20)

(21)

and a sliding mode controller is exploited for the control of proxy, i.e.,

τfb,SMC = Γ sgn(σ )

(22)

where Kp , Kd , Γ are all diagonal positive matrix, and Γ = diag{γ1 , γ2 , . . . , γn }. According to Newton’s second law, dynamics of proxy is mp¨ = τfb,PD − τfb,SMC

(23)

Considering that m = diag{0} and (20), (21), (22), it can be obtained from (23) that Kp ep + Kd e˙ p = Γ sgn(ψ − S e˙ p )

(28)

where Km is a diagonal positive matrix, stability analysis of (28) is established [38]. Although PBC can stabilize the closed-loop system of FJR, there are some drawbacks worthy for attention, listed as follows: 1. The useful non-collocated information is not directly used 2. There is no effective vibration suppression scheme for link side in control law In order to overcome the above-mentioned drawbacks, a proxy-based position controller with link side energy feedback is presented as follows:

τm = − KH H τc (φ ) + KH Hg(q¯ ) + τc (φ ) + u [ ] + Jq¯T Γ sat Γ −1 (Kd S −1 ψ + Kp ep ) − Km θ˙

(29)

H=

Subsequently, a PD controller is introduced to define the coupling between link and the proxy

τfb,PD = Kp ep + Kd e˙ p

τm = Jq¯T Γ sat[Γ −1 (Kd S −1 ψ + Kp ep )] + g(q¯ ) − Km θ˙

(17)

where S is a diagonal positive matrix. From (19), it is clear that eσ = eq¯ − ep , σ = ψ − S e˙ p

Based on the analysis above, the proxy-based controller (PBC) [38] is designed as

where KH is a positive constant to be designed later. H denotes the link side energy, defined as

In proxy-based control [38], it is assumed that there is a proxy located between q¯ and qd , whose position is p and mass m = diag{0}, and the following auxiliary variables are defined: ep = p − q¯ , eσ = qd − p, σ = eσ + S e˙ σ

(27)

(16)

It should be noted that the function sgn(z) is set-valued at z = 0, and the following relation between these two functions is established [57]:

∀ y, z ∈ R : y = sgn(z − y) ⇐⇒ y = sat(z)

(26)

and sat[Γ −1 (Kd S −1 ψ + Kp ep )] = sgn(eσ + S e˙ σ )

where

(25)

Proof of (25) can be obtained [38] and is omitted here. The meaning of (25) is that if proxy error ep moves along the trajectory (25), then (23) (with m = diag{0}) as well as the following two equalities hold: Kp ep + Kd e˙ p = Γ sgn(eσ + S e˙ σ )

(13)

3

(24)

1 T q˙ M(q)q˙ + Ut (q, θ ) − Ut (q¯ , θ ) (30) 2 Obviously, it can be concluded from property P1 and (12) that

∀q, q˙ , θ ∈ Rn : H ≥ 0 H = 0 ⇔ q˙ = 0, q = q¯

(31)

The additional term u = [u1 , u2 , . . . , un ]T , with individual element expressed as ui = −ρi θ˙i

n ∑

q˙ 2j

(32)

j=1

where ρi is a positive constant to be designed. Since ρi > 0, it is known that

θ˙ u = − T

n ∑ i=1

⎛ ρ θ˙ ⎝ 2 i i

n ∑

⎞ q2j

˙ ⎠≤0

(33)

j=1

Comparing (29) with (28), the controller proposed in this paper has made some changes. The first one is link side energy feedback: according to the expression of H, as long as there exists oscillatory behavior, it can be concluded that q˙ ̸ = 0 or q ̸ = q¯ . Therefore, H will get a certain value, and further the control input τm will drive the FJR system continuously until steady state. In this case, the controller designed in this paper is helpful to attenuate residual vibration. The second change is the additional term u: as will be explained in the subsequent section, it makes the system energy in the form of Lyapunov candidate function vanishes both on motor side and link side, which will also help to attenuate vibration. It is worth noting that ρi should not be very large in experiment configuration, since it will amplify the noise in velocity signal introduced by differential operation.

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L. Sun, W. Zhao, W. Yin et al. / Robotics and Autonomous Systems 121 (2019) 103272

4. Stability analysis In this section, the stability analysis will be derived with control input (29). Taking time derivative of H, and considering properties P1 and P2 as well as (9), it can be obtained that

˙ = −˙qT τfl (q, q˙ ) + θ˙ T τc (φ ) − θ˙ T g(q¯ ) H

(34)

For further stability analysis, the Lyapunov candidate function is chosen as 1 1 1 (35) V (t) = KH H 2 + θ˙ T J θ˙ + eTp Kp ep + ∥Γ eσ ∥1 2 2 2 where ∥ · ∥1 denotes the L1 norm. Obviously, V (t) is positive definite. Differentiating V (t) with respect to time yields V˙ (t) =θ˙ T [τm − τfm (θ , θ˙ ) − τc (φ )]

+

˙ + e˙ σ Γ sgn(eσ ) + KH H H˙ T

eTp Kp ep

Remark 1. It should be noted that V is a continuous but nonsmooth Lyapunov function, which is not differentiable at eσ = 0 because of the L1 norm. However, by referring to [58], the Lyapunov’s stability theorems can be applied by considering Clarke’s generalized gradient. In this paper, the generalized gradient of ∥Γ eσ ∥1 w.r.t. time is exactly e˙ Tσ Γ sgn(eσ ) since sgn(·) is a setvalued function defined in (16). For notational brevity, (d/dt)V (t) is expressed in terms of V˙ , which does not guarantee V˙ exists at eσ = 0, but the stability does hold. For further stability analysis, the following assumption should be introduced [38]: A3: Joint torque limits are sufficiently high so that the following inequality holds: Kd S −1 1ϵσ + Kp 1ϵp < γ

(36)

(44)

where γ = [γ1 , γ2 , . . . , γn ] and 1 = [1, 1, . . . , 1] , noting that the less-than sign works on individual element of the vector. T

By applying the control input from (29) to (36), it can be concluded that V˙ = − θ˙ T τfm (θ , θ˙ ) − θ˙ T KH H τc (φ ) + θ˙ T KH Hg(q¯ ) − θ˙ T Km θ˙ + θ˙ T u

[ ] + θ˙ T Jq¯T Γ sat Γ −1 (Kd S −1 ψ + Kp ep ) + eTp Kp e˙ p + e˙ Tσ Γ sgn(eσ ) + KH H H˙ (37)

T

Remark 2. In equilibrium point, the control input designed in this paper reduces to τm = g(q) + Jq¯T Γ sat[Γ −1 (Kd S −1 eσ + Kp ep )]. Therefore, assumption A3 is a necessary condition to make sure that joint torque will not be saturated.

Inserting the time derivative of H from (34) into (37) results in

To analyze the stability of equilibrium point, the region Ξ is defined as:

V˙ = − KH H q˙ T τfl (q, q˙ ) − θ˙ T τfm (θ, θ˙ ) − θ˙ T Km θ˙ + θ˙ T u

Ξ = {(p, q, θ, p˙ , q˙ , θ˙ )|V˙ = 0}

+ θ˙ T Jq¯T Γ sat Γ −1 (Kd S −1 ψ + Kp ep ) + e˙ Tp Kp ep + e˙ Tσ Γ sgn(eσ )

[

]

(38) Due to q˙¯ = Jq¯ (q¯ , θ )θ˙ from (13), with (27), Eq. (38) can be expressed as V˙ = − KH H q˙ T τfl (q, q˙ ) − θ˙ T τfm (θ, θ˙ ) − θ˙ T Km θ˙ + θ˙ T u eTp Kp ep

+ ˙

+ q˙¯ T Γ sgn(eσ + S e˙ σ ) + e˙ Tσ Γ sgn(eσ )

(39)

It is known from (19) that q˙¯ + e˙ p = −˙eσ . Considering (26), (39) can be further rewritten as V˙ = − KH H q˙ T τfl (q, q˙ ) − θ˙ T τfm (θ, θ˙ )

− θ˙ T Km θ˙ + θ˙ T u − e˙ Tp Kd e˙ p

(40)

− e˙ Tσ Γ sgn(eσ + S e˙ σ ) + e˙ Tσ Γ sgn(eσ ) ∀z , y ∈ Rn ∀X ∈ Dn yT X (sgn(z + y) − sgn(z)) ≥ 0

(41)

where Dn denotes the set of all n × n diagonal positive matrices, thus (40) can be finally expressed as

− e˙ Tp Kd e˙ p −

n ∑ i=1

ρi θ˙i2 ⎝

n ∑

⎞ q˙ 2j ⎠

(42)

j=1

It is known from (42) that V˙ is negative semi-definite, thus, V (t) ∈ ˙ ep and eσ are L∞ . Based on (35), it can be concluded that H , θ, all bounded. From (20), eq¯ is also bounded. That is, there exist positive constants ϵσ , ϵp , ϵq¯ such that

∥eσ ∥ < ϵσ , ∥ep ∥ < ϵp , ∥eq¯ ∥ < ϵq¯

q˙ = 0, θ˙ = 0, p˙ = q˙¯ , p˙ = 0 Therefore, q, p, q¯ , θ are all constants, and further q¨ = θ¨ = 0. Substituting these results into (1) yields g(q) = τc (φ ) KH H [τc (φ ) − g(q¯ )] = Jq¯T Γ sat[Γ −1 (Kd S −1 eσ + Kp ep )]

{

(43)

It is clear from (30) and H ∈ L∞ that q˙ ∈ L∞ and φ, (θ − q¯ ) ∈ L∞ , and these bounded terms can be utilized along with (14) to prove that Jq¯ ∈ L∞ ; q˙ , θ˙ ∈ L∞ along with (32) can be used to prove that u ∈ L∞ ; the boundedness of ep , eσ , eq¯ along with (19) can be used to conclude that p, q¯ ∈ L∞ . The above boundedness statements can be utilized along with (29) to prove that τm ∈ L∞ .

(46)

The unique solution for the first equation of (46) is q = q¯ . Substituting it into the second equation of (46) yields 0 = Jq¯T Γ sat[Γ −1 (Kd S −1 eσ + Kp ep )]

(47)

Namely, the following equation holds (48)

Since e˙ p = 0, it can be concluded from (26) and (27) that

Γ −1 Kp ep = sat(Γ −1 Kp ep + Γ −1 Kd S −1 eσ )

(49)

Combining with assumption A3, it yields

Γ −1 Kp ep = Γ −1 Kp ep + Γ −1 Kd S −1 eσ

V˙ ≤ − KH H q˙ T τfl (q, q˙ ) − θ˙ T τfm (θ, θ˙ ) − θ˙ T Km θ˙

⎛

In region Ξ , it can be concluded from (42) and assumption A2 that

Kd S −1 (qd − p) + Kp ep = 0

Combining (33) and the following inequality [57]:

(45)

(50)

Thus, p = qd . Substituting p = qd into (48) yields ep = 0, and further q = q¯ = qd . When q = qd , it is obvious that θ = θd with θd defined as

θd = qd + K −1 g(qd )

(51)

As a result, q = p = qd , θ = θd , p˙ = q˙ = θ˙ = 0 is the unique solution of V˙ = 0. By applying the nonsmooth version of La Salle’s invariance principle [58], the equilibrium point is globally asymptotically stable, that is, limt →∞ q(t) = qd and limt →∞ θ (t) = θd . The above analysis is summarized in the following theorem as the main result of this paper: Theorem 1. By properly choosing control gains, the desired state q = qd , θ = θd is globally asymptotically stable with control law (29), while all closed-loop signals remain bounded.

L. Sun, W. Zhao, W. Yin et al. / Robotics and Autonomous Systems 121 (2019) 103272

5

˙ and g¨ (q) are the first and second derivation of gravity, where g(q) respectively. In the experiment implementation, τfl (q, q˙ ) is simply neglected since it is small comparing with other terms; meanwhile, its exact model is also difficult to be obtained. Control gains of EGC are chosen as Kp = diag{32, 28}; Kd = diag{3, 1.8} (3) PID control [37], the control input is

τm = Kp (θd − θ ) − Kd θ˙ + g(qd ) + Ki

t

∫

[qd − q(η)]dη

(53)

0

Control gains of PID are chosen as Kp = diag{17, 18}, Ki = diag{8, 8}, Kd = diag{1.5, 1.8} Control parameters of proposed method are chosen as Kp = diag{55, 16}, Km = diag{3.2, 1.6} Kd = diag{4.3, 0.5}, S = diag{0.31, 0.13}

Γ = diag{16, 14}, ρ1 = ρ2 = 0.8, KH = 125 The gains of each control law are tuned repeatedly to make steady state error as small as possible. The following two cases are carried out to evaluate the performance of each control law:

Fig. 1. Self-built FJR platform.

5. Experiment validation To demonstrate the effectiveness of the proposed method, some experiments are carried out on a 2-DOF FJR platform shown in Fig. 1. The FJR is driven by Maxon DC servo motor with embedded coaxial encoders(4000PPR for first joint, 2000PPR for second joint). Link side position q is measured by Pepperl+Fuchs absolute encoder(65536PPR both), and velocity signals are obtained by numerical differential operation. NI9401 is utilized to read data of absolute encoders and convey them to NI 9039, while NI 9881 works for communication between NI 9039 and motor drivers. NI 9039 is used to generate real-time control commands for FJR and receive control algorithm programmed in PC using LabVIEW. Control period is 1 ms. Parameters identification of the FJR model is presented in [59]. Specifically, gravity and stiffness are detailed as follows: 6.17 sin(q1 ) + 0.53 sin(q1 + q2 ) 0.53 sin(q1 + q2 )

[ g(q) =

] (Nm)

K = diag{21.02, 17.75} (Nm/rad) The tolerant error e while calculating q¯ in (11) is designed as 0.001 (rad), and desired position of link side is qd = 0.6

[

]T

0.5

(rad)

In order to verify performance of the proposed method, the following three control laws are used as comparative approaches in the subsequent experiment: (1) Proxy-based control (PBC) [38], whose control law is detailed in (28), and control gains of PBC are chosen as Kp = diag{85, 45}, Kd = diag{2.8, 1.8} S = diag{0.2, 0.1}, Γ = diag{16, 10} Km = diag{3.6, 1.8} (2) PD with exact gravity cancellation (EGC) [36], the control law is

τm = τg + τ0 τg = g(q) + JK −1 g¨ (q) τ0 = Kp [qd − θ + K

−1

(52)

˙ ] g(q)] − Kd [θ˙ − K −1 g(q)

1. Nominal case, there is no external disturbance 2. External torque is applied to second link during transient state for a short while 5.1. Case 1: Nominal case Experiment results of this case are illustrated in Figs. 2–5. It is obvious that under the four control laws, both the first and second link can be driven to desired position with small steady state error. However, with EGC or PID, there exists obvious residual vibration in both two links, which leads to longer settling time and larger overshoot. The control performance of PBC is obviously better than above two algorithms, especially the smaller overshoot, but the vibration of first link still remains. Thanks to the LSE feedback, the method proposed in this paper can achieve better performance in terms of residual vibration suppression. From Figs. 4 and 5, it can be seen that control torques of the proposed method are significantly less than the other three methods, which implies lower power consumption of motion control is achieved. The detail performance indexes of the four control laws are collected in Tables 1 and 2, where Amp, eq1 , ts and µ denote the maximum residual vibration, steady state error, settling time and overshoot, respectively, and Amp is defined as half the difference between the first wave crest and the first wave trough. From Table 1, it can be seen that with the proposed method, all of the indexes are much smaller than the other three ones, especially in maximum residual vibration and overshoot. From Table 2, it can be seen that with the proposed method, although the settling time is a little longer than that of PBC, the steady state error and overshoot is much smaller. Table 1 Performance index of Case 1 (first joint). Indexes

Amp [rad]

eq1 [rad]

ts [s]

µ [%]

EGC PID PBC Proposed

0.167 0.102 0.014 0.004

0.011 0.007 0.004 0.001

1.917 2.410 0.706 0.472

40.577 37.437 2.181 0.005

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Fig. 5. Experiment result of Case 1: τm2 .

Fig. 2. Experiment result of Case 1: q1 .

Table 2 Performance index of Case 1 (second joint). Indexes

Amp [rad]

eq2 [rad]

ts [s]

µ [%]

EGC PID PBC Proposed

0.025 0.019 0.008 0.002

0.006 0.004 0.002 0.001

0.444 0.444 0.242 0.425

5.870 2.579 1.594 0.004

5.2. Case 2: External disturbance during transient state

Fig. 3. Experiment result of Case 1: q2 .

Experiment results of Case 2 are illustrated in Figs. 6–9, where the ‘‘disturbance’’ in Figs. 6 and 7 indicates that external torque is applied to the second link. It can be seen that as soon as the external torque vanishes, the link side is driven to move towards desired position. However, there exist large overshoots and severe oscillations under EGC and PID methods. Although the overshoot of PBC is smaller than above two methods, the residual vibration is more obvious than that of Case 1. However, The performance of the proposed method is almost unaffected by external disturbances. It is worthy for attention that there is obvious steady state error under EGC, since EGC ignores the effect of friction in both link side and motor side. Moreover, it can be seen from Figs. 6 and 7 that under PID method, the steady state error gradually decreases with time, which is caused by the integral term in the control input. Figs. 8 and 9 presents both τm1 and τm2 of PID also decrease with the time. However, the convergence rate of PID is slow and settling time is long. The detail information describing Case 2 is collected in Table 3. It can also be seen that the proposed method can achieve better performance than the other three ones, including smaller residual vibration, steady state error and overshoots. Table 3 Performance index of Case 2 (first joint).

Fig. 4. Experiment result of Case 1: τm1 .

Indexes

Amp [rad]

eq1 [rad]

µ [%]

EGC PID PBC Proposed

0.120 0.147 0.028 0.003

0.022 0.022 0.004 0.001

30.622 40.193 6.040 0.005

In summary, through the experiment results above as well as the expression of each control law, it can be seen that EGC is highly model-dependent, and the friction is ignored in both controller design and stability analysis. Via Cases 1 and 2, there exists obvious steady state error in EGC since it is difficult to

L. Sun, W. Zhao, W. Yin et al. / Robotics and Autonomous Systems 121 (2019) 103272

Fig. 6. Experiment result of Case 2: q1 .

Fig. 8. Experiment result of Case 2: τm1 .

Fig. 7. Experiment result of Case 2: q2 .

Fig. 9. Experiment result of Case 2: τm2 .

obtain exact model of FJR systems. Large Kp can attenuate steady state error, but it may also lead to larger overshoot and vibration. Although the damping injection strategy of PBC avoids large overshoot, there is no effective vibration suppression scheme. On the contrary, the proposed method introduces the link side energy, including potential and kinetic energy into feedback. Theoretically, if link side is shaking, then q˙ ̸ = 0 and q ̸ = q¯ , and the link side energy H as well as control input will get a certain value. As a result, the vibration of the link side can be effectively suppressed. Experimental results demonstrate the validity of the above theory, and the robustness against external disturbances is enhanced compared with PBC. Moreover, there is an additional term u in the control input (29); from (35) and (42), it can be concluded that with u, derivation of Lyapunov candidate function vanishes along both link and motor side variables, which is helpful to attenuate vibration of link side. Besides, the proposed

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method also has advantages in energy optimization since less control input is consumed in both two cases. It should be pointed out that the proposed method is an improvement based on PBC, thus large value of time constant matrix S will lead to damping action and the settling time will be a little long. 6. Conclusions The flexible actuation mostly intended for human–robot interaction has been developed rapidly in recent years. However, the position accuracy is greatly reduced due to flexible elements and the vibration is inspired easily. Thus, it is essential to develop vibration suppression strategy in the position control of FJR. In this paper, a proxy-based position controller is presented for FJR with link side energy feedback. By means of Lyapunov stability theorem and La Salle’s invariance principle, asymptotic stability

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of the closed-loop system and the boundedness of each signal are proved. Some comparing experiments are carried out on an FJR platform, which verify the vibration suppression ability and the robustness against external disturbances of the proposed method. In the future work, we plan to suppress vibration by trajectory planning of FJR, and further improve energy efficiency by exploring the energy storage characteristics of springs. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgment This work is supported in part by National Natural Science Foundation (NNSF) of China under Grant 61573198. References [1] A.D. Luca, Decoupling and feedback linearization of robots with mixed rigid/elastic joints, Internat. J. Robust Nonlinear Control 8 (11) (1998) 965–977. [2] M. Makarov, M. Grossard, P. Rodríguez-Ayerbe, D. Dumur, Modeling and preview H∞ control design for motion control of elastic-joint robots with uncertainties, IEEE Trans. Ind. Electron. 63 (10) (2016) 6429–6438. [3] K. Salisbury, B. Eberman, M. Levin, W. Townsend, The design and control of an experimental whole-arm manipulator, in: The Fifth International Symposium on Robotics Research, MIT Press, 1991, pp. 233–241. [4] S. Ozgoli, H. Taghirad, A survey on the control of flexible joint robots, Asian J. Control 8 (4) (2006) 332–344. [5] B. Siciliano, O. Khatib, Handbook of Robotics, Springer, 2016. [6] P. Tomei, A simple PD controller for robots with elastic joints, IEEE Trans. Automat. Control 36 (10) (1991) 1208–1213. [7] G.A. Pratt, M.M. Williamson, Series elastic actuators, in: International Conference on Intelligent Robots and Systems, IEEE, 1995, pp. 399–406. [8] B. Vanderborght, A. Albu-Schäffer, A. Bicchi, E. Burdet, D. Caldwell, R. Carloni, M. Catalano, G. Ganesh, M. Garabini, M. Grebenstein, et al., Variable impedance actuators: moving the robots of tomorrow, in: Intelligent Robots and Systems (IROS), 2012 IEEE/RSJ International Conference on, IEEE, 2012, pp. 5454–5455. [9] A.D. Wilson, J.A. Schultz, A.R. Ansari, T.D. Murphey, Dynamic task execution using active parameter identification with the Baxter research robot, IEEE Trans. Autom. Sci. Eng. 14 (1) (2017) 391–397. [10] M. Grebenstein, A. Albu-Schäffer, T. Bahls, M. Chalon, O. Eiberger, W. Friedl, R. Gruber, S. Haddadin, U. Hagn, R. Haslinger, et al., The DLR hand arm system, in: Robotics and Automation (ICRA), 2011 IEEE International Conference on, IEEE, 2011, pp. 3175–3182. [11] N. Roy, P. Newman, S. Srinivasa, CompActTM Arm: a Compliant Manipulator with Intrinsic Variable Physical Damping, MIT Press, 2013, p. 504. [12] N. Paine, J.S. Mehling, J. Holley, N.A. Radford, G. Johnson, C.-L. Fok, L. Sentis, Actuator control for the NASA-JSC valkyrie humanoid robot: a decoupled dynamics approach for torque control of series elastic robots, J. Field Robotics 32 (3) (2015) 378–396. [13] T.S. Tadele, T. de Vries, S. Stramigioli, The safety of domestic robotics: A survey of various safety-related publications, IEEE Robot. Autom. Mag. 21 (3) (2014) 134–142. [14] S. Haddadin, A. Albu-Schäffer, O. Eiberger, G. Hirzinger, New insights concerning intrinsic joint elasticity for safety, in: Intelligent Robots and Systems (IROS), 2010 IEEE/RSJ International Conference on, IEEE, 2010, pp. 2181–2187. [15] X. Li, Y. Pan, G. Chen, H. Yu, Adaptive human–robot interaction control for robots driven by series elastic actuators, IEEE Trans. Robot. 33 (1) (2017) 169–182. [16] Y. Fujimoto, T. Murakami, R. Oboe, Advanced motion control for nextgeneration industrial applications, IEEE Trans. Ind. Electron. 63 (3) (2016) 1886–1888. [17] E. Sariyildiz, H. Wang, H. Yu, A sliding mode controller design for the robust position control problem of series elastic actuators, in: Robotics and Automation (ICRA), 2017 IEEE International Conference on, IEEE, 2017, pp. 3055–3061.

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Lei Sun received the B. E. and M. E. degrees in Mechatronics Engineering from Tianjin University, China, in 1999 and 2002, respectively, and the Ph.D. degree in Control Theory and Control Engineering from Nankai University, China, in 2005. In 2005, he joined Nankai University, where he is currently an associate professor in the College of Artificial Intelligence. His research interest covers robotics and automatic control, teleoperation of robots in networks, wireless sensor network, micro-electro-mechanical systems (MEMS). Wen Zhao received the B. E. degree in Intelligent Science and Technology from Nankai University, Tianjin, China, in 2017. He is currently working toward the M. E. degree in Control Science and Engineering in the Institute of Robotics and Automatic Information System, Nankai University, Tianjin, China. His research interests include series elastic actuator and nonlinear control.

Wei Yin received the B. E. degree in Intelligent Science and Technology from Nankai University, Tianjin, China, in 2013, and the Ph.D. degree in Control Theory and Engineering in the Institute of Robotics and Automatic Information System of Nankai University, Tianjin, China, in 2018. He is currently working in the 14th Research Institute of China Electronics Technology Group Corporation. His research interests include motion planning and control of industrial robot and flexible joint robot. Ning Sun received the B.S. degree in measurement & control technology and instruments (with honors) from Wuhan University, Wuhan, China, in 2009, and the Ph.D. degree in control theory and control engineering (with honors) from Nankai University, Tianjin, China, in 2014. He is currently an Associate Professor with the Institute of Robotics and Automatic Information Systems, Nankai University, Tianjin, China. His research interests include cranes, wheeled robots, magnetic suspension systems, and nonlinear control with applications to mechatronic systems. Jingtai Liu received the B. E. and M. E. degree in Automation from Tianjin University, China, in 1983 and 1986, respectively, and the Ph.D. degree in Robotics from Nankai University, China, in 1998. In 1986, he joined Nankai University, where he is currently a professor in the College of Artificial Intelligence. His research interest covers robotics and automatic control, computer science and automatic information system.