AbstractâThis paper presents a biomimetic hybrid feedback feedforword (HFF) adaptive neural control for a class of robotic arms. The control structure includes ...
Biomimetic Hybrid Feedback Feedforword Adaptive Neural Control of Robotic Arms Yongping Pan, and Haoyong Yu Department of Biomedical Engineering, National University of Singapore, Singapore 117575 Email: {biepany; bieyhy}@ntu.edu.sg Abstract—This paper presents a biomimetic hybrid feedback feedforword (HFF) adaptive neural control for a class of robotic arms. The control structure includes a proportional-derivative feedback term and an adaptive neural network (NN) feedforword term, which mimics the human motor learning and control mechanism. Semiglobal asymptotic stability of the closed-loop system is established by the Lyapunov synthesis. The major difference of the proposed design from the traditional feedback adaptive approximation-based control (AAC) design is that only desired outputs, rather than both tracking errors and desired outputs, are applied as NN inputs. Such a slight difference leads to several attractive properties, including the convenient NN design, the decrease of the number of NN inputs, and semiglobal asymptotic stability dominated by control gains. Compared with previous HFF-AAC approaches, the proposed approach has two unique features: 1) all above attractive properties are achieved by a much simpler control scheme; 2) the bounds of plant uncertainties are not required to be known. Simulation results have verified the effectiveness and superiority of this approach. Keywords—Adaptive control, neural network, robotic manipulator, feedforword control, human motor learning control.
I.
I NTRODUCTION
Adaptive approximation-based control (AAC) using function approximators such as neural networks (NNs) and fuzzy logic systems (FLSs) has been demonstrated to be particularly suitable for nonlinear systems with nonparametric or even unknown uncertainties [1]–[14]. In traditional AAC approaches, approximators are applied in the feedback loop to compensate for some unknown functions depending on system states. Due to the local approximation property of most approximators and the feedback control structure, the traditional feedback AAC (FB-AAC) design faces some challenges in addressing the following problems [15]: 1) how to determine an approximation domain a priori so that approximators can be constructed; 2) how to ensure approximator inputs remain within the approximation domain so that function approximation is always valid; 3) how to reduce the number of approximator inputs so that implementation cost can be decreased. Hybrid feedback/feedforward (HFF) control that contains both feedback adjustment and NN feedforward compensation is promising for handling the aforementioned challenges. A key feature of this control strategy is that only desired outputs, rather than both tracking errors and desired outputs, are utilized as NN inputs. Since desired outputs can be prespecified, an approximation domain can be determined by the given desired outputs a priori so that the given desired outputs can remain within this domain doubtlessly. In addition, the number of NN inputs is decreased in this control strategy since tracking errors
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are not needed to be NN inputs. Yet at the early stage, most of the proposed HFF control approaches utilize nonadaptive NNs where tedious offline training should be involved [16]– [18]. These approaches are easy for implementation but lack of flexibility. Moreover, the establishment of the closed-loop stability is largely omitted in these approaches. By virtue of a second-order sliding-mode control technique [19], several HFF-AAC approaches were developed to achieve semiglobal asymptotic tracking results of some classes of uncertain nonlinear systems in [20]–[22], where NN feedforward is applied to redeem steady-state plant uncertainties, and robust integral of the sign of error feedback (RISE) is applied to cancel out transient-state plant uncertainties. Nevertheless, in these approaches, the increase of the plant orders for control synthesis leads to several limitations as follows: 1) the requirement of atypical smoothness on plant uncertainties and external disturbances limits the applicable range; 2) the requirement of known bounds on numerous user-defined uncertainties and their derivatives is impractical and leads to conservative design; 3) the control laws obtained are inevitably complex resulting in high implementation cost. On the other hand, other existing HFF-AAC approaches in [23]–[25] focus on the compensation of disturbances rather than of plant uncertainties. So far, there are only a few HFF-AAC approaches without the increase of the plant orders during control synthesis [26]– [28]. A HFF-AAC approach was proposed for a class of uncertain affine nonlinear systems with function-type control gains in [26], where the feedforward part contains two FLSs, the feedback part is constructed based on H ∞ control and nonlinear damping techniques, and a nonlinear damping term with known plant bounds is used for ensuring global stability. In [27], a HFF-AAC approach was presented for a class of affine nonlinear systems with constant control gains, where an adaptive bounding term with permanently positive estimation is utilized to ensure global stability. Regardless of the attractive features and great potentials, the approaches of [26], [27] are subject to some drawbacks as follows: 1) the stability results are dependent on a strict assumption that plant uncertainties are bounded by partially known functions; 2) favorable tracking performances are guaranteed at the cost of complex control laws with numerous design parameters; 3) there is a dilemma that exact analytical expressions of plant uncertainties are applied to derive partially known bounds in illustrative examples. Ishihara et al. [28] proposed an alternative HFF-AAC approach where dead-zone modification and protection ellipsoids are applied to establish closed-loop semiglobal uniformly bounded stability while preventing parameter drift. The major limitation of this approach is that 6 formulas with exact plant bounds are
applied to determine a feedback control gain. Motivated by the human motor learning and control mechanism [29]–[31], this paper presents a simple HFF-AAC scheme which is constituted by a proportional-derivative (PD) feedback term and a NN feedforward term for a class of robotic arms. The control law only contains two control terms with 4 design parameters. Semiglobal asymptotic stability of the closed-loop system under suitably large control gains is established by the Lyapunov synthesis. Compared with the existing HFF-AAC approaches in [26]–[28], the proposed approach possesses two unique features: 1) all challenges of the traditional FB-AAC design are completely resolved through a much simpler control scheme; 2) the bounds of plant uncertainties are not required to be known for control synthesis. This paper is organized as follows. The control problem is formulated in Section II. The HFF-AAC is developed in Section III. Simulation is given in Section IV. Finally, conclusions are summarized in Section V. Throughout this paper, R, R+ , Rn and Rn×m denote the spaces of real numbers, positive real numbers real n-vectors and real n × m-matrix, respectively, | · |, · and · F denote the absolute value, 2-norm and Frobenius norm, respectively, L∞ denotes the space of essentially bounded signals, tr(·) denotes the trace of a matrix, diag(·) denotes a diagonal matrix, min(·), max(·) and sup(·) represent the functions of minimum, maximum and supremum, respectively, and C k represents the space of functions whose k-order derivatives all exist and are continuous, where n, m and k are positive integers.
Define a position tracking error e1 and a filtered tracking error e2 as follows: e1 (t) := q d (t) − q(t) (2) e2 (t) := e˙ 1 (t) + α1 e1 (t) where α1 > 1/2 is a constant control gain parameter. Let e := [eT1 , eT2 ]T , xd := [q Td , q˙ Td , q¨Td ]T , q = [q1 , q2 , · · · , qn ]T , and q d = [qd1 , qd2 , · · · , qdn ]T . The control objective is to develop an efficient NN-based control strategy for the system (1) so that the system output q accurately tracks its desired signal q d for a wide range of initial states. III.
H YBRID C ONTROL D ESIGN
A. Feedback Feedforword Control Structure Differentiating e2 in (2) with respect to time t and multipling M to the both sides of the resulting equality yields ˙ − M (q)¨ q. M (q)e˙ 2 = M (q)(¨ q d + α1 e) Substituting the expression of M (q)¨ q in (1) into the above equality, one obtains ˙ q˙ M (q)e˙ 2 = M (q)(¨ q d + α1 e˙ 1 ) + C(q, q) ˙ − τ. + G(q) + F (q) After some manipulations, one gets to the open-loop filtered tracking error dynamics as follows: ˙ 2 + τd − τ M (q)e˙ 2 = H(xd , e) − C(q, q)e
(3)
with H(·) being a lumped plant uncertainty given by II.
P ROBLEM F ORMULATION
The class of robotic arms considered is described by the following Euler-Lagrange formulation [20]: ˙ q˙ + G(q) + F (q) ˙ = τ (t) M (q)¨ q + C(q, q)
(1)
˙ ∈ Rn×n where M (q) ∈ Rn×n is the inertia matrix, C(q, q) n is the centripetal-Coriolis matrix, G(q) ∈ R is the vector of ˙ ∈ Rn is the vector of frictions, τ (t) gravitational torques, F (q) n ˙ ∈ ∈ R denotes the vector of control torques, q(t) ∈ Rn , q(t) Rn and q¨ (t) ∈ Rn represent the vectors of positions, velocities and accelerations, respectively, and n is the system order. This study is based on the facts that q and q˙ are measurable, and ˙ G(q) and F (q) ˙ are unknown a priori. The M (q), C(q, q), following properties related to the system (1) are exploited in the subsequent development [20]. Property 1: The inertia matrix M (q) is symmetric and positive definite, and satisfies m0 ξ2 ≤ ξT M (q)ξ ≤ m(q) ¯ ξ2 , ∀ ξ ∈ Rn where m0 ∈ R+ is an unknown constant, and m(q) ¯ : Rn → R+ is an unknown function. ˙ is skew-symmProperty 2: The matrix M˙ (q) − 2C(q, q) ˙ (q) − 2C(q, q))ξ ˙ = 0, ∀ ξ ∈ Rn . etric such that ξ T (M ˙ G(q) Property 3: The plant functions M (q), C(q, q), ˙ are of class C 1 , ∀q ∈ Rn and ∀q˙ ∈ Rn . and F (q) Property 4: The vector of desired outputs q d (t) ∈ Rn (i) satisfies q d (t) ∈ L∞ for i = 0, 1, · · · , n + 1.
H(xd , e) = M (q)(¨ q d + α1 e˙ 1 ) ˙ q˙ d + α1 e1 ) + G(q) + F (q). ˙ + C(q, q)(
(4)
Let H(xd ) := H(xd , e)|e=0 be a steady-state lumped plant uncertainty. Using (4), one immediately obtains H(xd ) = M (q d )¨ q d + C(q d , q˙ d )q˙ d + G(q d ) + F (q˙ d ). (5) Subtracting and adding H(xd ) and the right side of (3) yields ˜ d , e) + H(xd ) − C(q, q)e ˙ 2−τ M (q)e˙ 2 = H(x
(6)
˜ d , e) := H(xd , e)−H(xd , 0). In a same manner in which H(x as the Remark 3 of [19], since H(·) in (5) is of C 1 , the Mean ˜ Value Theorem can be applied to H(·) to obtain ˜ d , e) ≤ ρ(e)e H(x
(7)
in which ρ(·) : R+ → R+ is a certain function that is globally invertible and strictly increasing. For the traditional AAC design, a NN with 5n inputs xd and e would be applied in the feedback loop to approximate H(xd , e) in (4). Yet here, a NN with only 3n inputs xd is applied in the feedforward loop to approximate H(xd ) in (5). The applied linearly parameterized NN is given by [1] ˆ d, W ˆ )=W ˆ T Φ(xd ) H(x
(8)
where Φ(·) : R3n → RN satisfying Φ(·) ≤ φ is the vector ˆ ∈ RN ×n is the matrix of adjustable of basis functions, W + weights, φ ∈ R is a certain constant, and N is the number of neurons. Note that to save space, more details on the NN
architecture is omitted and can be referred to Section 2 of [1]. Then, the control law is designed as follows: ˆ d, W ˆ) τ = α2 e2 + H(x (9)
Then, choose a Lyapunov function candidate
where α2 > 1/2 is a constant control gain parameter.
˜ F ]T ∈ R2n+1 for the closed-loop system with z := [eT , W constituted by (11) and (12). The following theorem is established to demonstrate the stability result of the closed-loop system under the proposed control scheme.
Define compact sets Ωd := {xd |xd ≤ cd } and Ωw := { ˆ 2 = tr(W ˆ TW ˆ ), and cd , cw ˆ |W ˆ F ≤ cw }, where W W F + ∈ R are prespecified constants. Next, define an optimal NN approximation error ε as follows: ˆ d, W ∗) (10) ε(xd ) := H d (xd ) − H(x where W ∗ is a matrix of optimal weights given by ˆ d, W ˆ )| . sup |H d (xd ) − H(x W ∗ := arg min ˆ W∈Ω w
xd ∈Ωd
˜ := W − W ˆ . Applying (9) and (10) to (6) leads to Let W the closed-loop filtered tracking error dynamics as follows: ∗
˜ +W ˜ T Φ(xd ) M (q)e˙ 2 = −α2 e2 + H ˙ 2 + ε(xd ). − C(q, q)e
(11)
Remark 1: The schematic diagrams of the proposed HFFAAC and the traditional FB-AAC are integrated in Fig. 1, where the former only includes the solid lines, and the later includes an additional dash line. It is worth noting that all NN inputs xd of the HFF-AAC are known before control, whereas 2n additional NN inputs e of the FB-AAC are unknown before control. Yet interestingly, such a slight difference leads to several attractive features as follows: 1) The domain of NN approximation can be determined a priori by the prespecified compact set Ωd ; 2) the number of NN inputs can be reduced from 5n to 3n resulting in a sharp decrease of computational cost; 3) semiglobal asymptotic stability can be guaranteed by the increase of control gains. Features 1 and 2 are straightforward, and feature 3 will be proven in Section III-B.
N ×N
1 T 1 1 ˜ T Γ−1 W ˜) e1 e1 + eT2 M (q)e2 + tr(W 2 2 2
(13)
Theorem 1: For the system (1) with Properties 1-4, the ˆ control law is designed as (9) with (8), (12) and W(0) ∈ Ωw , and the optimal NN approximation error ε in (11) is assumed to be 0 under a sufficiently large neuron number N . Then the closed-loop system achieves semiglobal asymptotic stability in the sense that the tracking error vector e converges to zero for an arbitrarily large domain of attraction S dominated by the PD control gains α1 and α2 . Proof: Differentiating V in (13) with respect to time t and using (2) and (11), one obtains V˙ = −α1 e1 2 + eT1 e2 + eT2 M (q)e˙ 2 ˙ (q)e2 /2 − tr(W ˆ˙ ) ˜ T Γ−1 W + eT2 M = −(α1 − 1/2)e12 − (α2 − 1/2)e22
˜ +δ+W ˜ T Φ(xd )) − tr(W ˜ T Γ−1 W ˆ˙ ) + eT2 (H ˙ (q) − 2C(q, q))e ˙ 2 /2. + eT (M 2
Noting Property 2, one immediately gets V˙ = −(α1 − 1/2)e1 2 − (α2 − 1/2)e22
ˆ˙ ). ˜ +δ+W ˜ T Φ(xd )) − tr(W ˜ T Γ−1 W + eT2 (H
(14)
ˆ˙ = ΓΦ(xd )eT in (12), one has For the case that W 2 ˜ T Φ(xd ). ˆ˙ ) = tr(W ˜ T Φ(xd )eT2 ) = eT2 W ˜ T Γ−1 W tr(W
B. Analysis of Closed-Loop Stability ˆ as follows: Design an adaptive law of W ˆ˙ = proj(ΓΦ(xd )eT ) W 2
V (z) =
(12)
denotes a constant, symmetric and posiin which Γ ∈ R tive definite matrix of learning rates, and proj(•) denotes a projection operator given by [1] ⎧ ˆ F < cw •, if W ⎪ ⎪ ⎪ ⎨ or W ˆ F = cw & tr(W ˆ T · •) ≤ 0; proj(•) = ˆ 2 , ˆW ˆ T · •/W ⎪ •−W ⎪ F ⎪ ⎩ ˆ F = cw & tr(W ˆ T · •) > 0. if W
Thus, according to the results of the projection operator in [1], ˆ (t) ∈ Ωw , ∀t ≥ 0 the adaptive law in (12) guarantees that: i) W T ˜ T ˆ˙ ) ≤ 0. ˜ T Γ−1 W ˆ if W (0) ∈ Ωw ; and ii) e2 W Φ(xd ) − tr(W Applying these results to (14) leads to 2 ˜ + δ). V˙ ≤ − (αi − 1/2)ei2 + eT2 (H (15) i=1
Applying ε = 0 and (7) to (15), one obtains V˙ ≤ −(ks − ρ(e))e2 with ks := min{α1 − 1/2, α2 − 1/2} ∈ R+ . Thus, one has V˙ ≤ −λc e2 , ∀ks > ρ(e) which is equivalent to V˙ ≤ −λc e2 , ∀e < ρ−1 (ks )
Fig. 1.
Schematic diagrams of HFF-AAC and FB-AAC.
(16)
where λc := ks − ρ(e) ∈ R+ . Then, the right part of (16) is utilized to determine a domain
(17) Dz := z ∈ R2n+1 |z < ρ−1 (ks ) .
Noting Property 1 and (16), define continuous positive-definite functions Ua , Ub and a continuous positive-semidefinite function U on z ∈ Dz as follows: ⎧ ⎨ Ua (z) := λa z2 Ub (z) := λb (z)z2 ⎩ U (z) := λ e2 c in which λa := min{m0 /2, λmin(Γ−1 )/2} ∈ R+ and λb (z) := max{m(q)/2, ¯ λmax (Γ−1 )/2} ∈ R+ . Then, V satisfies
Ua (z) ≤ V (z) ≤ Ub (z) V˙ (z) ≤ −U (z)
(18)
∀ t ≥ 0 and ∀ z ∈ Dz , which implies V (z) ∈ L∞ in Dz so ˜ ∈ L∞ in Dz . Combining with Properties 3 and 4, that e, W one knows that all terms at the right side of (11) are uniformly bounded in Dz , which implies e˙ 2 ∈ L∞ in Dz . Noting (2), one also has e˙ 1 ∈ L∞ in Dz . Thus, one gets e, e˙ ∈ L∞ in Dz . From e, e˙ ∈ L∞ in Dz and the definition of U (z), one gets U˙ (z) ∈ L∞ in Dz , which is a sufficient condition for U (z) being uniformly continuous in Dz . Now, since V satisfies (18), ∀ t ≥ 0 and ∀ z ∈ Dz , where Ua (z) and Ub (z) are continuous positive-definite and U (z) is uniformly continuous positivesemidefinite, the Invariance-Like Theorem (see Lemma 2 of [19]) can be invoked as in [20]–[22] to obtain limt→∞ U (z) = 0, ∀ z(0) ∈ S
(19)
Noting the definitions of U (z), one obtains limt→∞ e(t) = 0, ∀ z(0) ∈ S.
S IMULATION S TUDIES
Consider a two-link planar rigid robotic arm whose dynamic model is described by (1) with n = 2 and [32]
p1 + p2 + 2p3 cos q2 p2 + p3 cos q2 M (q) = , p2 + p3 cos q2 p2
−p3 q˙2 sin q2 −p3 (q˙1 + q˙2 ) sin q2 ˙ = C(q, q) , p2 q˙1 sin q2 0
p4 g cos q1 + p5 g cos(q1 + q2 ) G(q) = , p5 g cos(q1 + q2 ) ˙ ˙ = Kv q˙ + Kd tanh(q/) F (q) in which p := [p1 , p2 , · · · , p5 ]T ∈ R5 is a parameter vector related to the masses and lengths of links, Kv ∈ R2×2 is a diagonal coefficient matrix of viscous friction, Kd ∈ R2×2 is a diagonal coefficient matrix of dynamic friction, g ∈ R+ ˙ is the gravitational acceleration, tanh(q/) = [tanh(q˙1 /), tanh(q˙2 /)]T , tanh(·) is the hyperbolic tangent function, and ∈ R+ is a small constant. For simulation, set p = [2.90, 0.76, 0.87, 3.04, 0.87]T , Kv = diag(6, 2), Kd = diag(3, 1) and = 0.1. The control objective is to make q(t) track a fast varying desired output q d (t) = [(π/4) sin(πt), (π/4) cos(πt)]T . The design procedure of the control law (9) with (8) and (12) is as follows: Firstly, select Gaussian RBFs ⎧ j ⎨ μi = exp(−(qdi − (π/4)(j − 2))2 /2(π/6.4)2) μj = exp(−(q˙di − (π 2 /4)(j − 2))2 /2(π 2 /6.4)2 ) ⎩ i+2 μji+4 = exp(−(¨ qdi − (π 3 /4)(j − 2))2 /2(π 3 /6.4)2 ) with i = 1, 2 and j = 1, 2, 3 according to the information of q d such that the vector of basis functions Φ(xd ) perfectly covers the region of xd ; secondly, select α1 = 5, α1 = 100, ˆ (0) = 0 and Γ = 20000diag(1, · · · , 1). cw = 2000, W
where the domain of attraction S is given by
S := z ∈ Dz |Ub (z) < λa (ρ−1 (ks ))2 .
IV.
(20)
Moreover, because S in (19) can be arbitrarily enlarged by the increase of ks that is directly proportional to α1 and α2 to ˙ include any initial system states q(0) and q(0), the asymptotic stability result in (20) is semiglobal in this sense. Remark 2: A few HFF-AAC designs without the increase of the plant orders can be found in [26]–[28]. The control law of [26] has three control terms, i.e. a NN, a H ∞ control term and a nonlinear damping term, with two adaptive laws and over 15 design parameters. The control law of [27] has three control terms, i.e. a FLS, a PD control term and an adaptive bounding term, with three adaptive laws and more than 10 design parameters, where two adaptive laws are permanently positive. The control law of [28] utilizes 6 formulas with exact plant bounds to determine a feedback control gain. Differently, the control law (9) only has two control terms, i.e. a NN and a PD control term, with one adaptive law (12) and 4 design parameters. Consequently, it can be stated that the proposed approach completely resolves all challenges of the traditional FB-AAC design using a much simpler control scheme.
Simulations are carried out in Matlab 2013a environment running on Windows 7 and an Intel Core i5-3570 CPU. Let the solver be variable-step ode45 (Dormand-Prince), the running time be 10 s, the relative tolerance be 1 × 10−4 and other settings be defaults. The traditional FB-AAC shown in Fig. 1 is selected to be the compared approach. Let q(0) = [4π/9, ˙ 4π/9]T and q(0) = [0, 0]T so that they are outside of Ωd . Simulation trajectories of links 1 and 2 are shown in Fig. 2 and 3, respectively, where the trajectories of manipulator states and control torques are very similar for the two control laws. Comparison of tracking errors is given Fig. 4, where e1 = [e1 , e2 ]T , which demonstrates that the proposed HFF-AAC law achieves even smaller tracking errors than the traditional FB-AAC law although it has less NN inputs. Comparison of performance indexes is provided in Table I, which further validates higher tracking accuracy and much shorter simulation time of the proposed HFF-AAC law under similar control energy. TABLE I.
Controller type FB-AAC HFF-AAC a
C OMPARISON OF P ERFORMANCE I NDEXES FOR TWO D IFFERENT A PPROACHES
IAE1
Performance indexes IAE2 Ec1 Ec2
Tr (s)
1.6973 1.6249
0.7268 0.7102
86.42 31.41
29757 30717
3519 3624
IAE1 and IAE2 : integral absolute errors with respect to links 1 and 2, respectively; Ec1 and Ec2 : energies of τ1 and τ2 , respectively; Tr : real running time.
Link 1 position (°)
0.5 0 −0.5
1
2
3
4
5
6
7
8
9
2 0 −2 −4 0 200
1
2
3
4
5
6
7
8
9
0
−100
500 0
0.5 0 −0.5
0.1 2
0.15 3
τ1 by FB-AAC
0.2 4
5 time(s)
6
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8
9
10
2
3
4
10
−2
1
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3
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10
τ1 by HFF-AAC
100 0
500 0 0.1 2
0.15 3
0.2 4
5 time(s)
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8
1.5 Link 2 position (°)
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2 0 −2 −4 0 100
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10
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10
q˙d2 q˙2 by HFF-AAC
2 0 −2 −4 0 100
10
τ2 by FB-AAC
qd2 q2 by HFF-AAC
1
−1 0 4
10
q˙d2 q˙2 by FB-AAC
Link 2 control torque (N)
Link 2 position (°) Link 2 velocity (°/s)
9
0
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qd2 q2 by FB-AAC
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8
Simulation trajectories of the manipulator link 1. (a) Results by FB-ACC. (b) Results by HFF-ACC.
1
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τ2 by HFF-AAC
50 0 −50
200 0 −200
−100 0
0
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0.15 3
(a) Fig. 3.
7
(b)
1.5
−100 0
6
q˙d1 q˙1 by HFF-AAC
(a) Fig. 2.
5
2
−100
−500 0 0.05 −200 0 1
1
−4 0 200
10
100
qd1 q1 by HFF-AAC
1
−1 0 4
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q˙d1 q˙1 by FB-AAC
Link 1 velocity (°/s)
Link 1 velocity (°/s)
−1 0 4
Link 1 control torque (N)
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qd1 q1 by FB-AAC
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Link 1 control torque (N)
Link 1 position (°)
1.5
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(b)
Simulation trajectories of the manipulator link 2. (a) Results by FB-ACC. (b) Results by HFF-ACC.
V.
C ONCLUSIONS
This paper has established semiglobal asymptotic stability of uncertain robotic arms via a simple but efficient biomimetic HFF-AAC strategy. The salient feature of this approach is that it guarantees a minimum configuration of the control structure
and a minimum requirement of plant knowledge for the AAC design, which leads to a sharp decrease of implementation cost in terms of hardware selection, controller design and system debugging. Simulation results have verified the effectiveness and superiority of the proposed approach. In the further study,
0.3
0.1
Link 2 position errors (°)
−0.3
−0.6
0.02
−0.9 0 Zero line e1 by FB-AAC e1 by HFF-AAC
−1.2
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(a) Fig. 4.
ACKNOWLEDGMENT This work was supported in part by the Academic Research Fund Tier 1 (FRC) of the Ministry of Education, Singapore under WBS no. R-397-000-156-112, in part by the Seed Fund of the Engineering Design and Innovation Center, National University of Singapore, Singapore under Grant no. R-261503-002-133, and in part by the Science and Engineering Research Council, Agency for Science, Technology and Research (A*STAR), Singapore under Grant no. 1225100005. R EFERENCES
[2]
[3]
[4]
[5]
[6]
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Zero line e2 by FB-AAC e2 by HFF-AAC 1
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Zero line e˙ 2 by FB-AAC e˙ 2 by HFF-AAC
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(b)
Tracking error comparison for FB-AAC and HFF-AAC. (a) Results of manipulator link 1. (b) Results of manipulator link 2.
we will fucus on NN approximation-based biomimetic HFF variable impendence control of robotic arms.
[1]
−0.3
2.5
5
−1 0
−0.1
−0.7 0 3
0.1
Zero line e˙ 1 by FB-AAC e˙ 1 by HFF-AAC Link 1 velocity errors (°/s)
6 7
Link 2 velocity errors (°/s)
Link 1 position errors (°)
0
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