Adaptive Neural-Network Control for Redundant Nonholonomic Mobile Modular Manipulators Yangmin Li1 , Yugang Liu1 , and Shaoze Yan2 1 2 Faculty of Science and Technology, University of Macau, Av. Padre Tomas Pereira S.J., Taipa, Macao S.A.R., P.R. China {ymli,ya27401}@umac.mo Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, P.R. China yansz@tsinghua.edu.cn Abstract. This paper discusses the trajectory following issue for redundant nonholonomic mobile modular manipulators. Dynamic model is established and an adaptive neural-network controller is developed to control the end-effector to follow a desired spacial trajectory. The proposed algorithm doesn’t need any priori dynamics and provides a new solution for stabilization of redundant robotic selfmotions. Simulation results for a real robot demonstrate the proposed algorithm is effective. 1 Introduction A nonholonomic mobile modular manipulator can be defined as a kind of robot integrating a N -degree of freedom (DOF) modular manipulator together with a M -wheeled nonholonomic mobile platform. If the integrated structure has more DOFs than required, it is called a redundant one. This integration extends the workspace of the entire robot drastically. However, the nonholonomic constraints, the interactive motions, as well as the self-motions make the trajectory following task difficult to realize. Neural networks (NNs) with characteristics of not needing exact priori dynamic parameters and universal approximators are being widely used for robotic control. In related research work, back-propagation (BP) NN was used for vibration control of a 9-DOF redundant modular manipulator [1]. A multi-layer NN controller, which did not need off-line learning, was designed to control rigid robotic arms [2]. A fuzzyGaussian NN controller was proposed for trajectory tracking control of mobile robots [3]. A dual NN was presented for the bi-criteria kinematic control of redundant manipulators [4]. A sliding mode adaptive NN controller was devised for control of mobile modular manipulators [5]. In this paper, the dynamic model is established in Section 2. An adaptive NN controller (ANNC) is designed in task space in Section 3. Simulations are carried out in Section 4. Finally, some conclusions are given in Section 5. 2 Dynamic Modeling In this paper, a 3-wheeled nonholonomic mobile platform is studied as shown in Fig. 1(a), and only end-effector positions x = [px py pz ]T are concerned. The moJ. Wang, X. Liao, and Z. Yi (Eds.): ISNN 2005, LNCS 3498, pp. 271–276, 2005. c Springer-Verlag Berlin Heidelberg 2005
272 Yangmin Li, Yugang Liu, and Shaoze Yan bile platform is supposed to just move on a horizontal plane. The coordinate system can be defined as follows: an arbitrary inertial base frame OB XB YB ZB is fixed on the motion plane, while a frame Om Xm Ym Zm is attached to the mobile platform. The parameters can be observed in Fig. 1(b) in details. YB φL Ym Driving wheel G dm Driving wheel ⋯ Modular manipulator Xm φm Om ( xm , ym ) rf lG φR XB OB (a) A real robot (b) Platform motion Fig. 1. A real mobile modular manipulator and its motion on a plane T
Define ξ = xm ym φm φL φR , then the nonholonomic velocity constraints can be described as follows, [6]   rf · Cm /2 rf · Cm /2   dm Cm Sm − 2 −rf 0  rf · Sm /2 rf · Sm /2    dm   Cm Sm · rf /dm  = 0 (1)  −rf /dm 0 −r f 2   1 0 −Sm Cm 0 0 0 0 1 In short A (ξ) · S (ξ) = 0. T T

Define q = φL φR q1 · · · qN , and ζ = ξ T q1 · · · qN , then the Jacobian matrix can be derived by J= ∂x ∂x S (ξ) · S̄ (ξ) = T · 0N ×2 ∂ζ T ∂ζ 05×N IN ×N (2) The constrained dynamics can be determined by Eqn. 3, see [5] in details. H · ζ̈ + V · ζ̇ + G = B · τ + J T · Fext + C · λ (3) Where H, V and G denote the inertial matrix, the centripetal and coriolis matrix,
T T
and the gravitational force vector. B = 0n×3 In×n , C = A (ξ) 03×N , τ = T T
x
y z τL τR τ1 · · · τN , Fext = Fext is an external-force vector. Fext Fext From Eqn. 2 ẋ = J · q̇, ζ̇ = S̄ · q̇. (4) Solving Eqn. 4 and its derivative, yields ζ̇ = S̄J † ẋ + S̄ In − J † J q̇s

  ζ̈ = S̄ J † ẍ + In − J † J q̈s + S̄˙ − S̄J † J˙ J † ẋ + In − J † J q̇s (5) Adaptive Neural-Network Control 273 −1 Where J † = J T · J · J T is the Moore-Penrose generalized inverse of J; q̇s ∈ ℜn is an arbitrary joint velocity vector; J † · ẋ is a least-square solution; In − J † · J · q̇s ∈ ℵ (J), the null space of J, is a homogeneous solution. Define Jℵ ∈ ℜn×(n−m) as a matrix with all its columns being the normalized bases
T
 xTℵ . Substituting of ℵ (J), JE† = J † Jℵ , ẋℵ = JℵT · q̇s , and xE = xT Eqn. 5 into 3, and left multiplying JE† Where H̄ = JE† Ḡ = T JE† T T · S̄ T , yields H̄ · ẍE + V̄ · ẋE + Ḡ = τ̄ (6)
 T · S̄ T · H · S̄ · JE† , V̄ = JE† · S̄ T · H · S̄˙ − S̄ · J † · J˙ + V · S̄ , · S̄ T · G, τ̄ = JE† T · S̄ T · B · τ + J T · Fext ; JE† T · S̄ T · C · λ = 0. Remark 1. For any r ∈ ℜn , rT · H̄ · r ≥ 0. ˙ − 2V̄ · r = 0 . Remark 2. For any r ∈ ℜn , rT · H̄ Remark 3. If J is full rank, JE† is invertible, and JE = JE† −1
= JT Jℵ Remark 4. H̄, V̄ , Ḡ are all bounded as long as the Jacobian J is full rank. 3 T . Controller Design Let xd , ẋd and ẍd be the desired trajectory, velocity and acceleration in task space. The desired self-motions can be used to fulfil a secondary task. In this paper, the system is assumed to be far away from singularity, physical limits, obstacles. So, q̇sd and q̈sd  and  can be selected for the optimization problem of: min q̇ T · q̇ subject to ẋ = J · q̇. Then, xℵd , ẋℵd and ẍℵd can be determined. Let xEd = [ xTd | xTℵd ]T , then the error system can be defined by e (t) = xE (t) − xEd (t) , ẋr (t) = ẋEd (t) − Λ · e (t) , r (t) = ẋE (t) − ẋr (t) . (7) Where r (t) is the tracking error measure; Λ ∈ ℜn×n > 0 is a constant matrix. Substituting Eqn. 7 into 6, yields H̄ · ṙ (t) + V̄ · r (t) + H̄ · ẍr + V̄ · ẋr + Ḡ = τ̄ (8) It is verified that a multilayer perceptron (MLP) trained with BP algorithm can approximate any continuous multivariate functions to any desired degree of accuracy, provided that sufficient hidden neurons are available [7]. To ensure rapid convergence, multiple-input single-output (MISO) MLPs with only one hidden layer are applied in this paper as shown in Fig. 2(a). Output of this MLP is:    N Nh i   fN N (x, wih , who , θh , θo ) = xi · wihji + θhj · whoj + θo (9) ϕ j=1 i=1 Where ϕ (◦) is the activation function named hyperbolic tangent function; Ni and Nh represent neuron numbers; x is the inputs; wih , who , θh and θo denote weights and 274 Yangmin Li, Yugang Liu, and Shaoze Yan thresholds accordingly; the subscript ”i, h and o” represents the input, hidden and output layer respectively. T Define xin = [ ζ T q̇ T xTEd ẋTEd ẍTEd ] , h (xin ) = H̄ · ẍr + V̄ · ẋr + Ḡ. From Remark 4 and Eqn. 7, all the elements hk (k = 1, 2, · · · n) are bounded as long as J keeps full rank. Then, they can be approximated by MISO NNs, hk (xin ) = hN N k (xin ) + ǫk (xin ) (10) Where hN N k = fN N (xin , wkih , wkho , θkh , θko ); ǫk is the approximated error. Let ŵkih , ŵkho , θ̂kh and θ̂ko be estimates of wkih , wkho , θkh and θko respectively, define ĥN N k = fN N xin , ŵkih , ŵkho , θ̂kh , θ̂ko , then the Taylor series expansions of hN N k around ĥN N k can be derived N   N i h  ĥN N k ∂ ĥN N k 2 2 + ∂ ∂θ · w̃ + O w̃ · θ̃ko + O θ̃ko h̃N N k = kihji kihji ∂wkihji ko j=1 i=1 (11)  N h  ∂ ĥN N k ∂ ĥN N k 2 2 + ∂wkhoj · w̃khoj + ∂θkhj · θ̃khj + O w̃khoj + O θ̃khj j=1 Where h̃N N k = hN N k − ĥN N k , w̃kihji = wkihji − ŵkihji , w̃khoj = wkhoj − ŵkhoj , 2 2 2 2 , O w̃khoj , O θ̃khj and O θ̃ko θ̃khj = θkhj − θ̂khj , and θ̃ko = θko − θ̂ko ; O w̃kihji are higher-order terms. The ANNC is given by Eqn. 12, and the control system is shown in Fig. 2(b).  t   −1 T T τ = S̄ · B r (t) dt−Kǫ ·sgn (r) −J T ·Fext (12) ·JE · ĥN N −KP ·r−KI · 0 n Where ĥN N ∈ ℜ forms the adaptive NN term; KP , KI∈ ℜn×n are proportional  and integral gain matrices of the PID controller; Kǫ = diag kǫ1 kǫ2 · · · kǫn is the gain matrix of the robust term, and its elements are selected as follows: kǫk ≥ Nh   Ni  j=1 2 2 2 O w̃kihji + O θ̃khj + O w̃khoj i=1  2 + O θ̃ko + ǫk Substituting Eqn. 12 into 8, and considering 6 at the same time, yields  t H̄ · ṙ + V̄ · r + KP · r + KI · r (t) dt + Kǫ · sgn (r) + h̃N N + ǫ = 0 0 θ h1 ζ qɺ Wih xɺEd xEd θh2 x1 ⋮ ɺ xɺℵd yNN θh ( N h −1) ɺxɺd xℵd x Ni xd θ hN xɺℵd h xɺ d Input layer ( Hidden layer M u x M u x M u x θˆ ( ∂ f NN xin , wˆ ih , wˆ ho , θˆh ,θˆo x1 yNN ⋮ xNi NN θˆo Σ _ xɺEd + ih ho ( h ) o ∂f NN xin , wˆ ih , wˆ ho , θˆh , θˆo ) ) ∂ wˆ ih ) ∂ θˆh ∂ θˆo xE ζ qɺ e + Λ Σ in ,θˆ , θˆ Adaptive Law xE + _ ( ( x , wˆ , wˆ ∂f NN xin , wˆih , wˆ ho ,θˆh ,θˆo ∂ wˆ ho ∂f NN h ɺxɺEd xEd ) f NN xin , wˆ ih , wˆ ho , θˆh ,θˆo xin wˆ ih wˆ ho ⋮ θo ⋮ M u x ɺxɺEd Who xɺr Kε _ Σ + xɺE r KP ∫ _ _ Σ _ KI S  T −1 ⋅ B ⋅ JET + _ Σ Robot JT (a) A MISO NN xɺE Fext Output layer (b) An adaptive NN controller Fig. 2. A MISO NN and an adaptive NN controller (13) (14) Adaptive Neural-Network Control 275 Theorem 1. If KP > 0, KIT = KI > 0, the closed-loop system in Eqn. 14 is asymptotically stable under the adaptation laws given by Eqn. 15. The error signals are convergent with time, i.e., e (t) , ė (t) → 0, as t → +∞. ∂ ĥN N k ŵ˙ kihji = −Γwkihji · rk · ∂w , kihji ∂ ĥN N k ˙ ŵkhoj = −Γwkhoj · rk · ∂wkhoj , ˙ ĥN N k θ̂khj = −Γθkhj · rk · ∂∂θ , khj ˙ ∂ ĥN N k θ̂ko = −Γθko · rk · ∂θko . (15) Where Γwkihji , Γwkhoj , Γθkhj , and Γθko are positive constants. The terms with partial differentiation can be derived from Eqn. 10, details will not be listed here. Proof. Considering the following nonnegative Lyapunov candidate:   T  t t VS = 21 · rT · H̄ · r + 12 · 0 r (t) dt · KI · 0 r (t) dt  ! N Ni  w̃ 2 n 2 2 2 h  θ̃khj w̃khoj θ̃ko kihji 1  + 2· + Γw + Γθ ≥0 + Γθ Γw k=1 j=1 i=1 kihji khoj khj ko The time derivative of Lyapunov candidate is     n  t ˙ · r + 1 ·  θ̃ko ·θ̃˙ ko V̇S = rT · H̄ · ṙ + KI · 0 r (t) dt + 21 · rT · H̄ 2 Γθ k=1 ! ko   N Ni n h   ˙ kihji ˙ khoj w̃kihji ·w̃ θ̃ ·θ̃˙ khj w̃ ·w̃ + 12 · + khj + khoj Γw Γw Γθ k=1 j=1 i=1 kihji khoj (16) (17) khj ˙ ˙ ˙ ˙ Notice that w̃˙ kihji = −ŵ˙ kihji , w̃˙ khoj = −ŵ˙ khj , θ̃khj = −θ̂khj , θ̃ko = −θ̂ko . Substituting Eqn. 12 into 15, then substituting the result together with Eqns. 14,16 into 18, and considering Remark 2 at the same time, yields V̇S ≤ −rT · KP · r ≤ 0 (18) Therefore VS is a Lyapunov function, iff r = 0, VS and V̇S equal to zero. According to LaSalle’s theorem, the system is asymptotically stable and r → 0 as t → +∞. Define ℓp = {x (t) ∈ ℜn : x p < ∞} the p − norm space. From Eqns. 16 and 18, r (t) ∈ ℓ2 . According to Eqns. 7, e (t) ∈ ℓ2 ∩ℓ∞ , ė (t) ∈ ℓ2 , and e (t) → 0, as t → +∞. It is obvious that the higher-order terms in Eqn. 11 are bounded, so Kǫ ∈ ℓ∞ . Then, from Eqn. 14, ṙ (t) ∈ ℓ∞ . Since r (t) ∈ ℓ2 and ṙ (t) ∈ ℓ∞ , r (t) → 0 as t → +∞, which is followed by ė (t) → 0. End of proof. 4 Simulation Results The simulation is performed on a real robot as shown in Fig. 1(a), in which N = 3 and n = 5. Simulation time is 10 seconds and the end-effector is to follow the desired trajectory in Fig. 3(a). The gain matrices and constants are: Nh = 200, KP = diag {50}, KI = diag {10}, Γwkihji = Γwkhoj = Γθkhj = Γθko = 0.01, Kǫ = diag {50}, Λ = diag {2.0}. To examine the disturbance suppression ability of the ANNC, an exz = 1N is applied to the end-effector along the direction Zm at the time ternal force Fext instant t = 2s. 276 Yangmin Li, Yugang Liu, and Shaoze Yan The desired and the controlled locus are shown in Fig. 3(a). Figure 3(b) shows the self-motion velocities. The tracking position and velocity errors are given by Fig. 3(c)– 3(d). From these figures, we can see that the ANNC is effective to control the endeffector to follow a desired spacial trajectory. 1 1.5 3 1 2 0.5 1 pz (m) Desired 0 −1 • xℵ 1 0 −0.5 −1 x• ℵ2 Tracking velocity errors, m/s Controlled Tracing position erros, m Self−motion velocities, m/s 2 0.5 0 e z e x −0.5 e y −1 −1.5 −2 −1.5 1 • ez 0 • ex −1 • ey −2 −3 5 0 p (m) x −5 0 −5 5 p (m) y (a) Spacial locus −2 0 2 4 6 8 Time, s (b) Self-motions 10 −2 0 2 4 6 8 Time, s (c) Position errors 10 −4 0 2 4 6 8 10 Time, s (d) Velocity errors Fig. 3. Simulation results 5 Conclusions Dynamic model is established and an ANNC is developed for a general mobile modular manipulator. The proposed controller does not need precise knowledge of dynamic parameters in advance and can suppress bounded external disturbances effectively. The torque instability problem caused by self-motions of the redundant robot can be solved by the presented control algorithm. 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