Regulation of Robots. F.A. Dıaz-López. Higher Technological Institute of Huichapan Hidalgo, Mexico. L.E. Ramos Velasco. Polytechnic University of Pachuca.
Multiresolution Wavenet PID Control for Global Regulation of Robots F.A. D´ıaz-L´opez
L.E. Ramos Velasco
O.A. Dom´ınguez Ram´ırez
Higher Technological Institute of Huichapan Hidalgo, Mexico
Polytechnic University of Pachuca Hidalgo, Mexico
Center on Information Technology and Systems Hidalgo State University, Hidalgo, Mexico
V. Parra-Vega Robotics and Avanced Manufacturing Division, Research Center for Advanced Studies (Cinvestav), Mexico Abstract—A novel global PID control scheme for nonlinear MIMO systems is proposed and synthesized for a robot manipulator. Identification process is used for online tuning of the discrete linear PID feedback gains. Inverse dynamics identification is based on radial basis neural network with daughter RASP1 wavelets activation functions in cascaded with an infinite impulse response (IIR) filter in the output to prune irrelevant signals and nodes. The closed-loop system guarantees global regulation for a class of nonlinear dynamical systems, convenient for instance in plants whose dynamics are rather uncertain or unknown, such as in commercial robots. Real-time experimental study is carried out on a three degrees of freedom robotic haptic interface, the PHANToM Premium 1.0A. Results highlight the performance in global regulation with smooth control effort, without using the mathematical model of the robot, nor cumbersome tuning procedures typical of PID for robots.
I.
I NTRODUCTION
PID control, and some of its variants, are the main control scheme used in process control, [1]. Some causes that have contribute to this is its easiness of implementation, the independence of the model, the availability of explicit tuning procedures for the local linear approximation of the plant. There are even PID implemented in component-of-theshelf technological resources. However, the linear regulator PID algorithm presents limitations of stability, robustness and convergence when applied to nonlinear a processes [2], [3] and the tuning procedures can be very difficult, at best. Moreover, in particular, when PID is applied to nonlinear plants such as a robot, it requires explicit and difficult procedures to tune feedback gains and faces formidable challenges to show stability due to strong nonlinear couplings and common hard nonlinearities, like in practical setting of real robots. Then, similar to [4], we find of interest to obtain certain characteristics of global stability for a PID regulator of the nonlinear plant, while preserving the easiness of tuning or an online tuning procedure in place. Since classical PID schemes have constant positive definite feedback gains, defining a unique set of gains for the whole operational regime, even in a local domain, is a tantamount tasks. It is then natural to quest gain scheduling or time-varying feedback gains for different operational regimes. In [2], it is explored recent analytical and experimental techniques to tune gains, including auto-tuning or self-tuning, [5], [6]. In this realm, a recent and efficient scheme has been proposed using wavenet neural networksbased control (WNN) to identify the plant to tune accordingly online feedback gains. In this way, WNN is a different scheme
from classical neural network stream in that the function approximation capabilities of WNN is exploited for tuning feedback gains rather than implementing an inverse dynamics controller, such that the tuning scheme is designed to drive the error to zero. This requires less intensive computation effort since the aim is not function approximation but a quantitative objective measure of system deviation of origin. This stands for a compelling characteristic to introduce into a simple control, such a linear PID for a nonlinear plant. In this paper, we study a WNN-based PID scheme with an Infinite impulse response (IRR) filter is designed in the the output layer of the WNN as a pruning filter that further enhances a compact and efficient design to determine the useful signals that contribute to drive to zero the error. This stands for an effective alternative for global regulation of a nonlinear plant without any knowledge of the mathematical model nor its parameters, as well as its experimental verification on a robot manipulators. The proposed wavenet PID control is depicted in Figure I, where r(k) models white noise of zero mean of sensor measurements. The input of the identification scheme the control input is u(k), as well as the the persistence signal v(k), and the adaptation weights are driven by the identification error e(k) = y(k) − ˆy(k) the estimation error, for y(k) the plant output and ˆy(k) as its estimate. Notice that an IRR filter is at the output of this identification process, that delivers an estimation of the input ˆ matrix Γ(k). The input of the linear discrete controller is the tracking error (k) = ¯y(k)−yref (k), for yref (k) the reference and ¯y = y(k) + r(k).
𝒓(𝑘)
Sensors
a
𝒚𝒓𝒆𝒇 (𝑘)
+
-‐
𝜺(𝑘)
PID Controller
𝒑(𝑘)
𝒖(𝑘)
+
𝒅(𝑘)
𝒊(𝑘)
𝒚(𝑘)
Robot
-‐ 𝒆(𝑘) 𝒚(𝑘)
Auto-‐tuning PID
𝚪(𝑘)
Identification
𝒗(𝑘)
Fig. 1.
Block diagram of the proposed auto-tuning wavenet PID controller.
978-1-4673-5769-2/13/$31.00 ©2013 IEEE
II.
WAVENET I DENTIFICATION
A. Robot Dynamics Consider the nonlinear dynamic model of a rigid serial nlink robot manipulator as follows, in the continuous domain: H(q)¨ q + C(q, q) ˙ q˙ + g(q) = ν − νf + νd , n
scheme identifies approximately the inverse plant using as few neurons as possible to stands for an efficient approximator for practical applications due to its reduced computational load. The general interconnection and signal propagation is bl k . The IIR cascadpresented in Figure 2, where τl = ku(k)− al
(1)
n
where q ∈ R , q˙ ∈ R are the generalized position and n×n velocity joint coordinates, respectively, H(q) ∈ R denotes n×n a symmetric positive definite inertial matrix, C(q, q) ˙ ∈R n represents the Coriolis and centripetal forces, g(q) ∈ R n models the gravity forces, and ν ∈ R stands for the torque input. Term νf = B q+C ˙ tanh(Dq) ˙ stands for dissipative joint friction, for B, C, D positive definite n×n matrices modelling viscous damping and approximate linear dry friction and its coefficients, respectively. Disturbance torque νd is assumed a differentiable bounded time-varying function. Robot (1) can be represented as a general nonlinear MIMO dynamical system, whose general discrete state equation is, [7]: x(k + 1) = f [x(k), u(k), k] y(k) = g[x(k), k] n
(2) (3)
p
where x ∈ R , u, y ∈ R and f g
: :
Rn × Rp −→ Rn Rn −→ Rp
Fig. 2.
(4) (5)
are unknown smooth functions. Robot friction and disturbances are considered affine and state dependant, then those are represented in (4)-(5). Notice that input u(k) and system output y(k) are the only data available, and since the linearized system of (1) is observable around the equilibrium point, then there exists an input-output representation that can be reconstructed with a basis, [8]. Then, consider the following canonical realization: y(k + 1)
=
β[Y(k), U(k)]
Diagram of a wavenet neural network with an IIR filter in cascade.
ing recurrent structure plays a twofold role, firstly improving speed of learning by pruning those nodes with insignificant information from the cross contribution summation of daugthers wavelets, located in the third layer, see Figure 3. Notice the in there the forward delayed structure modulated by the input and the feedback loop and by the persistent signal to allow swapping a frequency range of interest. The mother
(6)
where Y(k) = [y(k) y(k − 1), · · · , y(k − n + 1)] U(k) = [u(k) u(k − 1), · · · , u(k − n + 1)]
(7) (8)
where vector field β maps the output y(k), input u(k) and their n − 1 past values in y(k + 1), [7]. In the seminal result of [9], it is established that a WNN can approximate smooth functions, then there exists a function βˆ that can be trained to converge at least locally to β. In this paper, we exploit this property of wavenets, or WNN, however and following [10], the estimation of system dynamics is used to tune feedback gains of a PID, not to implement an inverse dynamics scheme for linearization. Additionally, an IIR filter is designed in the output layer of the WNN to prune irrelevant signals to build an efficient identification scheme. B. Wavenet Identification Let a radial basis neural network scheme be used for the identification, where activation functions ψ(τ ) is a RASP1 daughter wavelet function ψj (τ ). Then, three IIR filter in cascade are introduced to filter neurons that have little or null contribution in the identification process, allowing a reduction in the number of iterations in the learning process, [11]. This
Fig. 3.
The IIR filter structure.
wavelet function ψ(k) generates daughter wavelets ψa,b (τ ) by its property of expansion or contraction and translation, [9]: 1 ψl (τl ) = √ ψ(τl ) a with a 6= 0; a, b ∈ R and Pp 2 1/2 τlj = /alj j=1 (uj − bl,j )
(9)
(10)
where j is the scale variable, alj allows expansion and contraction, and bl,j stands for the (l, j) translation variable at k,
in the classical role of a radial basis function (RBF), with the advantage of dealing with more refinement through daughters wavelets ψl (τl ). This last feature is essential in the present algorithm together with the pruning capability of the IIR filter. As suggested in [9], a wavelet RASP1 is a singularity-free normalization of the argument of the wavelet, RASP 1 =
(τ 2
τ + 1)2
Finally, the tuning update parameter becomes W(k + 1) = A(k + 1) = B(k + 1) = C(k + 1) = D(k + 1) =
Theorem 1: Let the system (2)-(3), and estimation (11). Then, cost function (13) is minimized and estimation error e(k) converges if (16) drives the online increment of parameters for µ chosen as follows
∂τ 1 3τ 2 − 1 = ∂bi,j a (τ 2 + 1)3 In this way, the i-th wavenet approximation signal with IIR filter can be calculated as: p X M X
ci,l zi (k − l)up (k) +
q=1 l=0
N X
di,j yˆi (k − j)v(k)
j=1
(11) where zi (k) =
L X
wi,l ψl (k)
0
