Obstacle Avoidance for Redundant Nonholonomic Mobile Modular ...
Obstacle Avoidance for Redundant Nonholonomic Mobile Modular Manipulators via Neural Fuzzy Approaches Yangmin Li and Yugang Liu Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Tom´ as Pereira S.J., Taipa, Macao S.A.R., P.R. China {ymli, ya27401}@umac.mo
Abstract. This paper addresses an obstacle avoidance issue for redundant nonholonomic mobile modular manipulators. On the basis of modular robot concept, an integrated dynamic modeling method is proposed, which takes both the mobile platform and the onboard modular manipulator as an integrated structure. A new obstacle avoidance algorithm is proposed which is mainly composed of two parts: a self-motion planner (SMP) and a robust adaptive neural fuzzy controller (RANFC). One important feature of this algorithm lies in that obstacles are avoided by online adjusting self-motions so that the end-effector task will not be affected unless the obstacles are just on the desired trajectory. The RANFC does not rely on exact aprior dynamic parameters and can suppress bounded external disturbance effectively. The effectiveness of the proposed algorithm is verified by simulations.
1
Introduction
In recent years, autonomous mobile manipulators have been paid extensively attention due to their wide applications. Compared with a conventional manipulator mounted on a fixed base, a mobile manipulator has much larger mobility in workspace. Modeling and control for redundant nonholonomic mobile modular manipulators are difficult to realize due to interactive motions, nonholonomic constraints and self-motions. Neural-fuzzy system has been widely used for robotic control due to its characteristics as universal approximators. Obstacle avoidance for mobile platforms [1,2] and robotic manipulators [3,4] is realized easily in a sense in case of treating respectively, complexity will increase significantly if obstacle avoidance is performed for an assembled mobile modular manipulator. Several obstacle avoidance schemes have been proposed, such as the artificial potential field (APF) method [1], the vector field histogram (VFH) algorithm [2], and the redundancy resolution scheme [3]. To get over the local minima problem of the APF method, harmonic artificial potential function is devised [4]. As for investigations on modular robots or mobile manipulators, back propagation neural network has been used for vibration control of redundant modular manipulators [5]. An integrated task planning and a decoupled L. Wang, K. Chen, and Y.S. Ong (Eds.): ICNC 2005, LNCS 3612, pp. 1109–1118, 2005. c Springer-Verlag Berlin Heidelberg 2005
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force/position control algorithm are proposed [6]. Regarding to researches on neural-fuzzy robotic control, a neural-fuzzy integrated controller is developed for mobile robot navigation and multi-robot convoying [7]. A robust adaptive generalized fuzzy-neural controller is presented for motion control of multi-link robotic manipulators [8]. Different from previous works in which the mobile platform and manipulator are modeled separately, the entire robot is modeled as an integrated structure in this paper. The redundancy of a redundant mobile manipulator is utilized to fulfil a task of obstacle avoidance by adjusting self-motions without changing end-effector’s specified job. Furthermore, most obstacle avoidance algorithms are only concerned with tasks of the start and goal points. However, the algorithm proposed can ensure the motion tasks not be affected during the entire course of navigation. In addition, unlike the previous schemes, this algorithm considers obstacles not only in the motion plane of the mobile platform, but also in the entire 3-D operational space of the onboard modular manipulator. This paper is arranged as follows. An integrated dynamic modeling method is proposed in Section 2. The obstacle avoidance algorithm is presented in Section 3. A simulation is carried out in Section 4. Conclusions are given in Section 5.
2
An Integrated Dynamic Modeling Method
In this paper, we analyze a 3-wheeled nonholonomic mobile modular manipulator, which is supposed to just move on a horizontal plane, as shown in Fig. 1. The coordinate systems are defined as follows: OB XB YB ZB forms an inertial base frame, and Om Xm Ym Zm is a frame fixed on the mobile platform. The origin of Om (xm , ym ) is selected as the midpoint of the line segment connecting the two fixed-wheel centers. Ym is along the line segment mentioned above. In Fig. 1(b), θi and ri are the yaw angle and steering radius at the time interval [ti , ti+1 ]. ∆SL , ∆SR , and ∆Sm represent advance of the left wheel, the right wheel, and Om respectively. φL and φR are rotating angles of the left and right front wheels, φr and βr denote rotating angles of the castor wheel around its own axis and the fixed bar. dm , rf , lG , lr , dr and rr are all constants determined by physical structures. Assume ∆t = ti+1 − ti → 0, from Fig. 1(b) ⎧ m rf ·Cm ⎪ x˙ m = lim ∆x = 2 · φ˙ L + φ˙ R ⎪ ∆t ⎪ ∆t→0 ⎨ r ·S m = f 2 m · φ˙ L + φ˙ R y˙ m = lim ∆y (1) ∆t ∆t→0 ⎪ ⎪ ⎪ r ∆φ f m ⎩ φ˙ m = lim = − dm · φ˙ L − φ˙ R ∆t ∆t→0
Where Sm = sin (φm ), Cm = cos (φm ). In the same way, we have ˙ φ˙ r = Crm ·x˙ m +Srmr·ry˙ m −lr ·Sr ·φm ˙ β˙ r = Srm ·x˙ m −Crm ·y˙ m −(dr −lr ·Cr )·φm dr
(2)
Redundant Nonholonomic Mobile Modular Manipulators YB
1111
Xm
θi
φL
Ym
Omi +1 ( xmi +1 , ymi +1 )
ri
Driving Wheel
φR
φr
rr
Xm
G
rf lr
∆S m
∆S R
φ mi
dm lG
Castor Wheel
βr
dr
∆S L
Ym
φmi +1
G
Omi ( xmi , ymi )
Driving Wheel
XB
OB
(a) Prototype
(b) Coordinate system
Fig. 1. Prototype and coordinate system for a mobile modular manipulator
Where Srm = sin (βr + φm ) , Crm = cos (βr + φm ), Sr = sin (βr ) , Cr = cos (βr ). Define ξ = [xm ym φm φr βr φL φR ]T , then from Eqs. 1 and 2, the nonholonomic constraints can be given by A (ξ) · S (ξ) = 0.
(3)
Where the matrices A (ξ) and S (ξ) can be detailed by ⎡
Cm Cm −Sm ⎢ Sm Sm Cm ⎢ d d ⎢− m m 0 ⎢ 2 2 A=⎢ 0 0 ⎢ 0 ⎢ 0 0 0 ⎢ ⎣ −rf 0 0 0 −rf 0
⎡ rf ·Cm ⎤T Crm −Srm 2 rf ·Sm ⎢ Srm Crm ⎥ ⎢ 2r ⎥ f ⎢ − dm −lr Sr kr ⎥ ⎢ ⎥ ⎢ r (d C +2l m r r Sr ) f −rr 0 ⎥ ⎥ ,S=⎢ 2dm rr ⎢ 0 dr ⎥ ⎢ rf (dm Sr +2kr ) ⎥ ⎢ 2dm dr 0 0 ⎦ ⎣ 1 0 0 0
rf ·Cm 2 rf ·Sm r2
⎤
⎥ ⎥ ⎥ ⎥ dm rf (dm Cr −2lr Sr ) ⎥ ⎥ 2dm rr ⎥ rf (dm Sr −2kr ) ⎥ ⎥ 2dm dr ⎦ 0 f
1
(4) Where kr = dr − lr Cr . According to modular robot concept, the mobile platform can be treated as a special module attached to the base of the modular manipulator. From DenavitHartenberg notation, transformation matrix of the ith module with respect to OB XB YB ZB can be derived, see [9] for details. Let ζ = [ξ T q1 · · · qn ]T , q = [φL φR q1 · · · qn ]T , x = [px py pz ]T , then ∂x ∂ζ ∂x S 07×n ζ˙ = · q˙ = · · q˙ (5) · q, ˙ x˙ = 0n×2 In×n ∂q ∂ζ ∂q In short ζ˙ = S · q, ˙ x˙ = J · q. ˙ In this paper, as long as n > 1, the robot will be redundant, then from Eq. 5 (6) q˙ = J † · x˙ + In+2 − J † · J · q˙s
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−1 Where J † = J T · J · J T is the Moore-Penrose generalized inverse of J, q˙s ∈ n+2 is an arbitrary vector. Let Jℵ ∈ (n+2)×(n−1) be a matrix with all its columns be the normalized bases of ℵ (J), which is the null space of J. Then J · Jℵ = 03×(n−1) , JℵT · J † = 0(n−1)×3 , JℵT · Jℵ = I(n−1) , Jℵ · JℵT = In+2 − J † · J.
(7)
T Define x˙ ℵ = JℵT · q˙s , xE = xT xTℵ , JE† = J † Jℵ . From Eqs. 6 and 7, we have ζ˙ = S¯ · JE† · x˙ E , ζ¨ = S¯ · JE† · x ¨E + S¯˙ · JE† − S¯ · J † · J˙ · x˙ E (8) The constrained dynamics can be determined by [10] M · ζ¨ + V · ζ˙ + G = B · τ + J T · Fext + C · λ
(9)
T T Where B = 0(n+2)×5 In+2 , C = A 05×n , Fext is an external force T T vector, λ = λ1 · · · λ5 are Lagrange multipliers, τ = τL τR τ1 · · · τn are corresponding driving torques. T Substituting Eq. 8 into Eq. 9, and left multiplying JE† · S¯T , yields ¯ · x¨E + V¯ · x˙ E + G ¯ = τ¯ M
(10)
˙ † ¯ † ,G ¯ − M SJ ¯= ¯ † , V¯ = J † T S¯T M SJ ¯ † J˙ + V SJ ¯ = J † T S¯T M SJ Where M E E E E E † T T † T T T † JE S¯ G, τ¯ = JE S¯ B τ + J T Fext ; and JE S¯T Cλ = 0 is eliminated. Remark 1. The following properties hold for Eq. 10: 1) For any r ∈ n+2 , rT · ˙ ¯ · r ≥ 0; 2) For any r ∈ n+2 , rT · M ¯ − 2V¯ · r = 0; 3) If J is full rank, M −1 T ¯ , V¯ , G ¯ ∈ ∞ . Here ∞ = = J T Jℵ ; 4) If J is full rank, M JE = JE† n {x (t) ∈ : x∞ < ∞}.
3 3.1
A New Obstacle Avoidance Algorithm Problem Formulation
According to whether on the desired end-effector trajectory or not, obstacles can be divided into two kinds: the task-consistent one and the task-inconsistent one, see Fig. 2(a). The task-consistent obstacles can be avoided by on-line adjusting self-motions. However, the task-inconsistent obstacles can not be avoided without affecting end-effector executed tasks. One solution to avoid task-inconsistent obstacles is to regenerate the desired end-effector task, which belongs to the high-level decision making problem and is beyond the discussion of this paper. In this paper, only task-consistent obstacles are concerned and redundancy of the robot is supposed to be high enough to avoid obstacles just by adjusting selfmotions. Obstacle avoidance is realized online, so the exact positions of obstacles
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need not be known in advance. Furthermore, the mobile modular manipulator is supposed to work in a unstructured environment and the obstacles can be detected by sonar, infrared, laser range finder, vision or some other sensors in a realtime manner. task-inconsistent obstacle desired path
M
x1
Π
w1 task-consistent obstacle j
dij
M
x2
M
w2
Σ wNr f NFS
M
critical point i
xNi −1
M
Π 1 1
Σ 1
xN i
M
(a) Obstacle classification
Π
(b) A MISO NFS
Fig. 2. Obstacle classification and a MISO neural-fuzzy system
3.2
Self-motion Planning
Let xℵd , x˙ ℵd and x ¨ℵd be the desired self-motions. Assume the system is far away from singularity and physical limits, then the self-motions can be used specially for obstacle avoidance. If a point on the robot gets too close to an obstacle (dij < dc ), this point can be called a critical point, and dc is called the cut-off distance. The artificial potential function for the ith critical point and the j th obstacle can be defined by 2 1 1 1 · k · − φij (q) = 2 φ dij dc , dij < dc (11) 0, dij ≥ dc Where kφ > 0 is a constant coefficient, dij = xci − xoj is the nearest distance between the ith critical point and the j th obstacle, as shown in Fig. 2(a). Here, xci = [pcix pciy pciz ]T and xoj = [pojx pojy pojz ]T are position vectors for the ith critical point and the j th obstacle with respect to OB XB YB ZB . To avoid obstacles in a realtime manner, the self-motions can be planned to optimize the following function: Φ (q) =
No Nc j=1 i=1
φij (q)
(12)
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Where No and Nc are the numbers of obstacles and critical points respectively. Then No Nc ∂φij (q) ∂Φ (q) q˙sd = − =− (13) ∂q ∂q j=1 i=1 Where
∂φij (q) ∂q
can be derived from Eq. 12.
⎧ 1 ∂φij (q) ⎨ − kφ · dij − = ⎩ ∂q
1 dc
dT ij dij 3
·
·
∂xci ∂qT
−
∂xoj ∂qT
T
0,
, dij < dc dij ≥ dc (14)
Then, xℵd , x˙ ℵd and x ¨ℵd can be determined. 3.3
Robust Adaptive Neural-fuzzy Controller Design
Theorem 1. (Universal Approximation Theorem [12]) The multiple inputs single output (MISO) fuzzy logic system (FLS) with center average defuzzifier, product inference rule and singleton fuzzifier, and Gaussian membership function can uniformly approximate any nonlinear functions over a compact set U ∈ n to any degree of accuracy. If the FLS described above is realized by a neural network (NN), a neural fuzzy system (NFS) can be obtained as shown in Fig. 2(b). Output of this NFS is given by 2 N N r i x − wj · exp − i σji ji fN F S =
j=1
i=1
Nr
N i
j=1
i=1
2 xi −ji exp − σji
(15)
th
Where xi is the i input variable, wj denotes the j th centroids for the output fuzzy sets, i = 1, 2, · · · , Ni , j = 1, 2, · · · , Nr , here Ni and Nr represent the number of input variables and rules respectively. ji and σji are the mean and standard derivation of the Gaussian membership functions accordingly. Let xd , x˙ d and x¨d be desired task-space position, velocity and acceleration. Define xEd = [ xTd | xTℵd ]T , then the error system can be defined as e (t) = xE (t) − xEd (t) , x˙ s (t) = x˙ Ed (t) − Λ · e (t) , s (t) = x˙ E (t) − x˙ s (t) (16) Where s (t) is the tracking error measure, Λ is a constant positive definite matrix. Substituting Eq. 16 into 10, yields ¯ · s˙ (t) + V¯ · s (t) + M ¯ ·x ¯ = τ¯ M ¨s (t) + V¯ · x˙ s (t) + G (17) ˙ x˙ s , x ¯ ·x ¯ According to the universal apDefine h ζ, ζ, ¨s + V¯ · x˙ s + G. ¨s = M proximation theorem mentioned above and Remark1, each element of h can be approximated by a MISO NFS as long as the Jacobian matrix J is full rank. Then
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h = hN F S + . Here hN F S ∈ n+2 are NFS approximates of h and are approximated errors, hN F Sk = fN F S (xin , k , σk , wk ), k , σk ∈ Nr ×Ni , and wk ∈ Nr are adjustable parameter matrices for these NFS, xin = [ζ T q˙T xTEd x˙ TEd x¨TEd ]T are corresponding inputs, k = 1, 2, · · · , n + 2. ¯ , then the adjustable parameters can be Assume xin ∈ [xin , x ¯in ], h ∈ h, h initialized by kji0 = xini + j ·
x ¯ini −xini , Nr
σkji0 =
x ¯ini −xini , Nr
wkji0 = hk + j ·
¯ k −h h k Nr
(18)
Let ˆ k, σ ˆk , and w ˆk be estimates of k , σk and wk respectively. Taking ˆ N F Sk = fN F S (xin , ˆ k, σ ˆk , w ˆk ), the Taylor series expansions of hN F Sk around h yields Nr Ni ˆ ∂ ˆh ∂ hN F Sk ∂ˆ hN F Sk N F Sk ˜ N F Sk = · ˜ kji + ·˜ σkji + ·w ˜kj +hres (19) h ∂ kji ∂σkji ∂wkj j=1 i=1 ˜ N F Sk = hN F Sk − h ˆ N F Sk , Where h ˜ = kji − ˆ kji , σ ˜kji = σkji − σ ˆkji and N kji N i 2 2 2 2 r O ˜ kji +O σ ˜kj ; here O ˜kji +O w w ˜kj = wkj −w ˆkj ; hres = ˜ kji , 2 2 j=1 i=1 O σ ˜kji , and O w ˜kj are higher-order terms. The RANFC is represented by t −1 T JE ˆ s (t) dt − J T Fext (20) hN F S − K sgn (s) − KP s (t) − KI τ = S¯T B 0
KIT
= KI > 0 are proportional and integral gain matrices, Where KP > 0, K = diag { k1 · · · k(n+2) } is the gain matrix for the robust term, with its elements determined by kk ≥ |k | + |hres |. Substituting Eq. 20 into Eq. 17 and considering Eq. 10, yields the following error equation t ˜ N F S = 0 (21) ¯ · s˙ (t) + V¯ · s (t) + + KI · M s (t) dt + KP · s (t) + K · sgn (s) + h 0
Theorem 2. The closed-loop system in Eq. (21) is asymptotically stable under the adaptation laws given by Eq. (22). The error signals are convergent along with time, i.e., e (t) , e˙ (t) → 0, as t → +∞. ∂ˆ hN F Sk ˙ ∂ˆ hN F Sk ˙ ∂ ˆhN F Sk ,σ ˆ kji = −Γσkji sk ,w ˆ kj = −Γwkj sk . ˆ˙ kji = −Γkji sk ∂ kji ∂σkji ∂wkj (22) ˆ N F Sk ∂ h ˆ N F Sk ˆ N F Sk ∂h ∂h Where ∂kji , ∂σkji , and ∂wkj can be detailed by N i
ˆ N F Sk ∂h ∂wkj
=
x − ˆ kji 2 exp − iniσ ˆ
ˆ N F Sk ∂h , ∂kji x − N ˆ N F Sk r N i ˆ kji 2 ∂h exp − iniσ ˆ kji ∂σkji j=1 i=1 i=1
kji
= =
ˆ N F Sk ) ˆ 2(xini − ˆ kji )(w ˆkj −h ∂ hN F Sk 2 ∂wkj σ ˆkji ˆ N F Sk (xini − ˆ kji ) ∂ h · ∂kji σ ˆkji
(23)
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Proof. Considering the following Lyapunov candidate: ¯ · s + 1 · t s (t) dt T · KI · t s (t) dt · sT · M 2 0 0 N 2 Nr n+2 2 2 i ˜ kji σ ˜kji w ˜ kj + 12 · + + ≥0 Γ Γσ Γw
VS =
1 2
k=1
j=1 i=1
kji
kji
kj
The time derivative of Lyapunov candidate is ˙ ¯ ¯ · s˙ + KI · t s (t) dt + sT ·M·s V˙ S = sT · M 2 0 N Nr n+2 i ˙ kj ˜ kji · ˜˙ kji σ ˜ kji ·σ ˜˙ kji w ˜kj ·w ˜ + Γw + + Γσ Γ k=1
j=1
i=1
kji
kji
(24)
(25)
kj
From Eq. 21, we have ¯ · s˙ + KI · t s (t) dt = −sT · V¯ · s sT · M 0 ˜ N F S + k · sgn (s) + −sT · KP · s − sT · h
(26)
ˆ˙ kji , σ ˜˙ kji = −σ ˆ˙ kji , w ˜˙ kj = −w ˆ˙ kj . Substituting Eqs. Notice that ˜˙ kji = − 19,22 and 26 into 25, and considering kk ≥ |k | + |hres | at the same time, yields V˙ S ≤ −sT · KP · s ≤ 0
(27)
From Eqs. 24 and 27, VS is a Lyapunov function. According to LaSalle’s theorem, the system is asymptotically stable and s → 0 as t → +∞. Define
p = {x (t) ∈ n : xp < ∞}. From Eqs. 24 and 27, s (t) ∈ 2 . According to Eq. 16, e (t) ∈ 2 ∩ ∞ , e˙ (t) ∈ 2 , and e (t) → 0, as t → +∞. Since V˙ S ≤ 0, ˜ kji , σ ˜kji , w ˜kj ∈ ∞ , if the Jacobian is full rank, VS ∈ ∞ , which implies that ˆ N F S ∈ ∞ . hN F S ∈ ∞ and kji , σkji , wkj ∈ ∞ , so, ˆ kji , σ ˆkji , w ˆkj ∈ ∞ and h Considering that hres ∈ ∞ , so K ∈ ∞ . Then, from Eq. 21, s˙ (t) ∈ ∞ . Since s (t) ∈ 2 and s˙ (t) ∈ ∞ , s (t) → 0 as t → +∞, which is followed by e˙ (t) → 0. End of the proof.
4
Simulation Results
The simulation is performed on a real robot composed of a 3-wheeled mobile platform and a 4-DOF modular manipulator, as shown in Fig. 1(a). In order to verify the algorithm, the robot is required to follow a spacial trajectory in Fig. 3(a), which has been planned to ensure the robot far away form singularities or joint limits. Two ball-like task-consistent obstacles with radius of 0.2m are considered, one is on the motion plane of the mobile platform and the other is on the way of the modular manipulator as shown in Fig. 3(b). The simulation time is selected as 20 seconds. Each element of h is approximated by a NFS. The gain matrices and constants are selected as follows: KP = diag {100} , KI = diag {10} , K = diag {50} , Γkji = 0.1, Γσkji =
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0.8 0.75 0.7
z
p ,m
0.65 0.6 0.55 0.5 0.45
Controlled
0.4 2
Desired
1.5
1.5 1
1 0.5
0.5 p ,m
0
0
y
−0.5 −0.5
−1
p ,m x
(a) Desired and controlled locus
(b) Obstacle avoidance
Fig. 3. Desired and control locus and obstacle avoidance
0.4
0.2
0.3
0.15
0.1
0.2
Tracking velocity errors, m/s
Tracking position errors, m
ey 0.05
ex
0
−0.05 e
z
−0.1
e•
y
0.1 e•
0
x
−0.1 •
ez −0.2
−0.15
−0.3
−0.2
−0.25
−0.4 0
2
4
6
8
10
12
14
16
18
Time, s
(a) Tracking position errors
20
0
2
4
6
8
10
12
14
16
18
20
Time, s
(b) Tracking velocity errors
Fig. 4. Tracking position and velocity errors
0.1, Γwkj = 0.1, Λ = diag {2.0} and Nr = 200. The cut-off distance is selected as dc = 0.5m, and the coefficient is determined by kφ = 1.0. The desired and the controlled locus are shown in Fig. 3(a). Two obstacles are avoided by controlling self-motions of the mobile modular manipulator in Figure 3(b). The tracking position and velocity errors are given by Fig. 4. It can be observed that the proposed algorithm is effective in both avoiding obstacles and controling the end-effector to follow a desired spacial trajectory simultaneously.
5
Conclusions
A mobile modular manipulator composed by a 3-wheeled nonholonomic mobile platform and a n-DOF onboard modular manipulator is investigated in this paper. Firstly, an integrated dynamic modeling method is presented. Secondly, a new obstacle avoidance algorithm using self-motions is proposed, which can avoid obstacles without affecting the end-effector planning task. Lastly, simulations are performed on a real mobile modular manipulator, which demonstrate that the proposed algorithm is effective. The dynamic modeling method and the obstacle
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avoidance algorithm proposed can be extended to study other kinds of mobile manipulators as well.
Acknowledgements This work was supported by the Research Committee of University of Macau under grant RG082/04-05S/LYM/FST.
References 1. Ge, S. S. and Cui, Y. J.: New Potential Functions for Mobile Robot Path Planning. IEEE Trans. on Robotics and Automation. 16 (2000) 615–620 2. Borenstein, J. and Koren, Y.: The Vector Field Histogram-fast Obstacle Avoidance for Mobile Robots. IEEE Trans. on Robotics and Automation. 7 (1991) 278–288 3. Zhang, Y. N. and Wang, J.: Obstacle Avoidance for Kinematically Redundant Manipulators Using a Dual Neural Network. IEEE Trans. on System, Man, and Cybernetics-Part B: Cybernetics. 34 (2004) 752–759 4. Kim, J. O. and Khosla, P. K.: Real-Time Obstacle Avoidance Using Harmonic Potential Functions. IEEE Trans. on Robotics and Automation. 8 (1992) 338–349 5. Li, Y., Liu, Y., Liu, X. and Peng, Z.: Parameters Identification and Vibration Control in Modular Manipulators. IEEE/ASME Trans. Mechatronics. 9 (2004) 700–705 6. Tan, J., Xi, N. and Wang, Y.: Integrated Task Planning and Control for Mobile Manipulators. Int. Journal of Robotics Research. 22 (2003) 337–354 7. Ng, K. C. and Trivedi, M. M.: A Neuro-Fuzzy Controller for Mobile Robot Navigation and Multirobot Convoying. IEEE Trans. on Syst., Man, and Cybern.- Part B: Cybernetics. 28 (1998) 829–840 8. Er, M. J. and Gao, Y.: Robust Adaptive Control of Robot Manipulators Using Generalized Fuzzy Neural Networks. IEEE Trans. on Industrial Electronics. 50 (2003) 620–628 9. Li, Y. and Liu, Y.: Control of a Mobile Modular Manipulator Moving on a Slope. IEEE Int. Conf. on Mechatronics, Turkey (2004) 135–140 10. de Wit, C. C., Siciliano, B. and Bastin, G.: Theory of Robot Control. SpringerVerlag London Limited (1996) 11. Haykin, S.: Neural Networks: A Comprehensive Foundation. 2nd edition. PrenticeHall, Inc. (1999) 12. Wang, L. X., Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice-Hall International, Inc., (1994).
Abstract. This paper addresses an obstacle avoidance issue for redundant nonholonomic mobile modular manipulators. On the basis of modular robot concept, an integrated dynamic modeling method is proposed, which takes both the mobile platform and the onboard modular manipulator as an integrated structure. A new obstacle avoidance algorithm is proposed which is mainly composed of two parts: a self-motion planner (SMP) and a robust adaptive neural fuzzy controller (RANFC). One important feature of this algorithm lies in that obstacles are avoided by online adjusting self-motions so that the end-effector task will not be affected unless the obstacles are just on the desired trajectory. The RANFC does not rely on exact aprior dynamic parameters and can suppress bounded external disturbance effectively. The effectiveness of the proposed algorithm is verified by simulations.
1
Introduction
In recent years, autonomous mobile manipulators have been paid extensively attention due to their wide applications. Compared with a conventional manipulator mounted on a fixed base, a mobile manipulator has much larger mobility in workspace. Modeling and control for redundant nonholonomic mobile modular manipulators are difficult to realize due to interactive motions, nonholonomic constraints and self-motions. Neural-fuzzy system has been widely used for robotic control due to its characteristics as universal approximators. Obstacle avoidance for mobile platforms [1,2] and robotic manipulators [3,4] is realized easily in a sense in case of treating respectively, complexity will increase significantly if obstacle avoidance is performed for an assembled mobile modular manipulator. Several obstacle avoidance schemes have been proposed, such as the artificial potential field (APF) method [1], the vector field histogram (VFH) algorithm [2], and the redundancy resolution scheme [3]. To get over the local minima problem of the APF method, harmonic artificial potential function is devised [4]. As for investigations on modular robots or mobile manipulators, back propagation neural network has been used for vibration control of redundant modular manipulators [5]. An integrated task planning and a decoupled L. Wang, K. Chen, and Y.S. Ong (Eds.): ICNC 2005, LNCS 3612, pp. 1109–1118, 2005. c Springer-Verlag Berlin Heidelberg 2005
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force/position control algorithm are proposed [6]. Regarding to researches on neural-fuzzy robotic control, a neural-fuzzy integrated controller is developed for mobile robot navigation and multi-robot convoying [7]. A robust adaptive generalized fuzzy-neural controller is presented for motion control of multi-link robotic manipulators [8]. Different from previous works in which the mobile platform and manipulator are modeled separately, the entire robot is modeled as an integrated structure in this paper. The redundancy of a redundant mobile manipulator is utilized to fulfil a task of obstacle avoidance by adjusting self-motions without changing end-effector’s specified job. Furthermore, most obstacle avoidance algorithms are only concerned with tasks of the start and goal points. However, the algorithm proposed can ensure the motion tasks not be affected during the entire course of navigation. In addition, unlike the previous schemes, this algorithm considers obstacles not only in the motion plane of the mobile platform, but also in the entire 3-D operational space of the onboard modular manipulator. This paper is arranged as follows. An integrated dynamic modeling method is proposed in Section 2. The obstacle avoidance algorithm is presented in Section 3. A simulation is carried out in Section 4. Conclusions are given in Section 5.
2
An Integrated Dynamic Modeling Method
In this paper, we analyze a 3-wheeled nonholonomic mobile modular manipulator, which is supposed to just move on a horizontal plane, as shown in Fig. 1. The coordinate systems are defined as follows: OB XB YB ZB forms an inertial base frame, and Om Xm Ym Zm is a frame fixed on the mobile platform. The origin of Om (xm , ym ) is selected as the midpoint of the line segment connecting the two fixed-wheel centers. Ym is along the line segment mentioned above. In Fig. 1(b), θi and ri are the yaw angle and steering radius at the time interval [ti , ti+1 ]. ∆SL , ∆SR , and ∆Sm represent advance of the left wheel, the right wheel, and Om respectively. φL and φR are rotating angles of the left and right front wheels, φr and βr denote rotating angles of the castor wheel around its own axis and the fixed bar. dm , rf , lG , lr , dr and rr are all constants determined by physical structures. Assume ∆t = ti+1 − ti → 0, from Fig. 1(b) ⎧ m rf ·Cm ⎪ x˙ m = lim ∆x = 2 · φ˙ L + φ˙ R ⎪ ∆t ⎪ ∆t→0 ⎨ r ·S m = f 2 m · φ˙ L + φ˙ R y˙ m = lim ∆y (1) ∆t ∆t→0 ⎪ ⎪ ⎪ r ∆φ f m ⎩ φ˙ m = lim = − dm · φ˙ L − φ˙ R ∆t ∆t→0
Where Sm = sin (φm ), Cm = cos (φm ). In the same way, we have ˙ φ˙ r = Crm ·x˙ m +Srmr·ry˙ m −lr ·Sr ·φm ˙ β˙ r = Srm ·x˙ m −Crm ·y˙ m −(dr −lr ·Cr )·φm dr
(2)
Redundant Nonholonomic Mobile Modular Manipulators YB
1111
Xm
θi
φL
Ym
Omi +1 ( xmi +1 , ymi +1 )
ri
Driving Wheel
φR
φr
rr
Xm
G
rf lr
∆S m
∆S R
φ mi
dm lG
Castor Wheel
βr
dr
∆S L
Ym
φmi +1
G
Omi ( xmi , ymi )
Driving Wheel
XB
OB
(a) Prototype
(b) Coordinate system
Fig. 1. Prototype and coordinate system for a mobile modular manipulator
Where Srm = sin (βr + φm ) , Crm = cos (βr + φm ), Sr = sin (βr ) , Cr = cos (βr ). Define ξ = [xm ym φm φr βr φL φR ]T , then from Eqs. 1 and 2, the nonholonomic constraints can be given by A (ξ) · S (ξ) = 0.
(3)
Where the matrices A (ξ) and S (ξ) can be detailed by ⎡
Cm Cm −Sm ⎢ Sm Sm Cm ⎢ d d ⎢− m m 0 ⎢ 2 2 A=⎢ 0 0 ⎢ 0 ⎢ 0 0 0 ⎢ ⎣ −rf 0 0 0 −rf 0
⎡ rf ·Cm ⎤T Crm −Srm 2 rf ·Sm ⎢ Srm Crm ⎥ ⎢ 2r ⎥ f ⎢ − dm −lr Sr kr ⎥ ⎢ ⎥ ⎢ r (d C +2l m r r Sr ) f −rr 0 ⎥ ⎥ ,S=⎢ 2dm rr ⎢ 0 dr ⎥ ⎢ rf (dm Sr +2kr ) ⎥ ⎢ 2dm dr 0 0 ⎦ ⎣ 1 0 0 0
rf ·Cm 2 rf ·Sm r2
⎤
⎥ ⎥ ⎥ ⎥ dm rf (dm Cr −2lr Sr ) ⎥ ⎥ 2dm rr ⎥ rf (dm Sr −2kr ) ⎥ ⎥ 2dm dr ⎦ 0 f
1
(4) Where kr = dr − lr Cr . According to modular robot concept, the mobile platform can be treated as a special module attached to the base of the modular manipulator. From DenavitHartenberg notation, transformation matrix of the ith module with respect to OB XB YB ZB can be derived, see [9] for details. Let ζ = [ξ T q1 · · · qn ]T , q = [φL φR q1 · · · qn ]T , x = [px py pz ]T , then ∂x ∂ζ ∂x S 07×n ζ˙ = · q˙ = · · q˙ (5) · q, ˙ x˙ = 0n×2 In×n ∂q ∂ζ ∂q In short ζ˙ = S · q, ˙ x˙ = J · q. ˙ In this paper, as long as n > 1, the robot will be redundant, then from Eq. 5 (6) q˙ = J † · x˙ + In+2 − J † · J · q˙s
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−1 Where J † = J T · J · J T is the Moore-Penrose generalized inverse of J, q˙s ∈ n+2 is an arbitrary vector. Let Jℵ ∈ (n+2)×(n−1) be a matrix with all its columns be the normalized bases of ℵ (J), which is the null space of J. Then J · Jℵ = 03×(n−1) , JℵT · J † = 0(n−1)×3 , JℵT · Jℵ = I(n−1) , Jℵ · JℵT = In+2 − J † · J.
(7)
T Define x˙ ℵ = JℵT · q˙s , xE = xT xTℵ , JE† = J † Jℵ . From Eqs. 6 and 7, we have ζ˙ = S¯ · JE† · x˙ E , ζ¨ = S¯ · JE† · x ¨E + S¯˙ · JE† − S¯ · J † · J˙ · x˙ E (8) The constrained dynamics can be determined by [10] M · ζ¨ + V · ζ˙ + G = B · τ + J T · Fext + C · λ
(9)
T T Where B = 0(n+2)×5 In+2 , C = A 05×n , Fext is an external force T T vector, λ = λ1 · · · λ5 are Lagrange multipliers, τ = τL τR τ1 · · · τn are corresponding driving torques. T Substituting Eq. 8 into Eq. 9, and left multiplying JE† · S¯T , yields ¯ · x¨E + V¯ · x˙ E + G ¯ = τ¯ M
(10)
˙ † ¯ † ,G ¯ − M SJ ¯= ¯ † , V¯ = J † T S¯T M SJ ¯ † J˙ + V SJ ¯ = J † T S¯T M SJ Where M E E E E E † T T † T T T † JE S¯ G, τ¯ = JE S¯ B τ + J T Fext ; and JE S¯T Cλ = 0 is eliminated. Remark 1. The following properties hold for Eq. 10: 1) For any r ∈ n+2 , rT · ˙ ¯ · r ≥ 0; 2) For any r ∈ n+2 , rT · M ¯ − 2V¯ · r = 0; 3) If J is full rank, M −1 T ¯ , V¯ , G ¯ ∈ ∞ . Here ∞ = = J T Jℵ ; 4) If J is full rank, M JE = JE† n {x (t) ∈ : x∞ < ∞}.
3 3.1
A New Obstacle Avoidance Algorithm Problem Formulation
According to whether on the desired end-effector trajectory or not, obstacles can be divided into two kinds: the task-consistent one and the task-inconsistent one, see Fig. 2(a). The task-consistent obstacles can be avoided by on-line adjusting self-motions. However, the task-inconsistent obstacles can not be avoided without affecting end-effector executed tasks. One solution to avoid task-inconsistent obstacles is to regenerate the desired end-effector task, which belongs to the high-level decision making problem and is beyond the discussion of this paper. In this paper, only task-consistent obstacles are concerned and redundancy of the robot is supposed to be high enough to avoid obstacles just by adjusting selfmotions. Obstacle avoidance is realized online, so the exact positions of obstacles
Redundant Nonholonomic Mobile Modular Manipulators
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need not be known in advance. Furthermore, the mobile modular manipulator is supposed to work in a unstructured environment and the obstacles can be detected by sonar, infrared, laser range finder, vision or some other sensors in a realtime manner. task-inconsistent obstacle desired path
M
x1
Π
w1 task-consistent obstacle j
dij
M
x2
M
w2
Σ wNr f NFS
M
critical point i
xNi −1
M
Π 1 1
Σ 1
xN i
M
(a) Obstacle classification
Π
(b) A MISO NFS
Fig. 2. Obstacle classification and a MISO neural-fuzzy system
3.2
Self-motion Planning
Let xℵd , x˙ ℵd and x ¨ℵd be the desired self-motions. Assume the system is far away from singularity and physical limits, then the self-motions can be used specially for obstacle avoidance. If a point on the robot gets too close to an obstacle (dij < dc ), this point can be called a critical point, and dc is called the cut-off distance. The artificial potential function for the ith critical point and the j th obstacle can be defined by 2 1 1 1 · k · − φij (q) = 2 φ dij dc , dij < dc (11) 0, dij ≥ dc Where kφ > 0 is a constant coefficient, dij = xci − xoj is the nearest distance between the ith critical point and the j th obstacle, as shown in Fig. 2(a). Here, xci = [pcix pciy pciz ]T and xoj = [pojx pojy pojz ]T are position vectors for the ith critical point and the j th obstacle with respect to OB XB YB ZB . To avoid obstacles in a realtime manner, the self-motions can be planned to optimize the following function: Φ (q) =
No Nc j=1 i=1
φij (q)
(12)
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Where No and Nc are the numbers of obstacles and critical points respectively. Then No Nc ∂φij (q) ∂Φ (q) q˙sd = − =− (13) ∂q ∂q j=1 i=1 Where
∂φij (q) ∂q
can be derived from Eq. 12.
⎧ 1 ∂φij (q) ⎨ − kφ · dij − = ⎩ ∂q
1 dc
dT ij dij 3
·
·
∂xci ∂qT
−
∂xoj ∂qT
T
0,
, dij < dc dij ≥ dc (14)
Then, xℵd , x˙ ℵd and x ¨ℵd can be determined. 3.3
Robust Adaptive Neural-fuzzy Controller Design
Theorem 1. (Universal Approximation Theorem [12]) The multiple inputs single output (MISO) fuzzy logic system (FLS) with center average defuzzifier, product inference rule and singleton fuzzifier, and Gaussian membership function can uniformly approximate any nonlinear functions over a compact set U ∈ n to any degree of accuracy. If the FLS described above is realized by a neural network (NN), a neural fuzzy system (NFS) can be obtained as shown in Fig. 2(b). Output of this NFS is given by 2 N N r i x − wj · exp − i σji ji fN F S =
j=1
i=1
Nr
N i
j=1
i=1
2 xi −ji exp − σji
(15)
th
Where xi is the i input variable, wj denotes the j th centroids for the output fuzzy sets, i = 1, 2, · · · , Ni , j = 1, 2, · · · , Nr , here Ni and Nr represent the number of input variables and rules respectively. ji and σji are the mean and standard derivation of the Gaussian membership functions accordingly. Let xd , x˙ d and x¨d be desired task-space position, velocity and acceleration. Define xEd = [ xTd | xTℵd ]T , then the error system can be defined as e (t) = xE (t) − xEd (t) , x˙ s (t) = x˙ Ed (t) − Λ · e (t) , s (t) = x˙ E (t) − x˙ s (t) (16) Where s (t) is the tracking error measure, Λ is a constant positive definite matrix. Substituting Eq. 16 into 10, yields ¯ · s˙ (t) + V¯ · s (t) + M ¯ ·x ¯ = τ¯ M ¨s (t) + V¯ · x˙ s (t) + G (17) ˙ x˙ s , x ¯ ·x ¯ According to the universal apDefine h ζ, ζ, ¨s + V¯ · x˙ s + G. ¨s = M proximation theorem mentioned above and Remark1, each element of h can be approximated by a MISO NFS as long as the Jacobian matrix J is full rank. Then
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h = hN F S + . Here hN F S ∈ n+2 are NFS approximates of h and are approximated errors, hN F Sk = fN F S (xin , k , σk , wk ), k , σk ∈ Nr ×Ni , and wk ∈ Nr are adjustable parameter matrices for these NFS, xin = [ζ T q˙T xTEd x˙ TEd x¨TEd ]T are corresponding inputs, k = 1, 2, · · · , n + 2. ¯ , then the adjustable parameters can be Assume xin ∈ [xin , x ¯in ], h ∈ h, h initialized by kji0 = xini + j ·
x ¯ini −xini , Nr
σkji0 =
x ¯ini −xini , Nr
wkji0 = hk + j ·
¯ k −h h k Nr
(18)
Let ˆ k, σ ˆk , and w ˆk be estimates of k , σk and wk respectively. Taking ˆ N F Sk = fN F S (xin , ˆ k, σ ˆk , w ˆk ), the Taylor series expansions of hN F Sk around h yields Nr Ni ˆ ∂ ˆh ∂ hN F Sk ∂ˆ hN F Sk N F Sk ˜ N F Sk = · ˜ kji + ·˜ σkji + ·w ˜kj +hres (19) h ∂ kji ∂σkji ∂wkj j=1 i=1 ˜ N F Sk = hN F Sk − h ˆ N F Sk , Where h ˜ = kji − ˆ kji , σ ˜kji = σkji − σ ˆkji and N kji N i 2 2 2 2 r O ˜ kji +O σ ˜kj ; here O ˜kji +O w w ˜kj = wkj −w ˆkj ; hres = ˜ kji , 2 2 j=1 i=1 O σ ˜kji , and O w ˜kj are higher-order terms. The RANFC is represented by t −1 T JE ˆ s (t) dt − J T Fext (20) hN F S − K sgn (s) − KP s (t) − KI τ = S¯T B 0
KIT
= KI > 0 are proportional and integral gain matrices, Where KP > 0, K = diag { k1 · · · k(n+2) } is the gain matrix for the robust term, with its elements determined by kk ≥ |k | + |hres |. Substituting Eq. 20 into Eq. 17 and considering Eq. 10, yields the following error equation t ˜ N F S = 0 (21) ¯ · s˙ (t) + V¯ · s (t) + + KI · M s (t) dt + KP · s (t) + K · sgn (s) + h 0
Theorem 2. The closed-loop system in Eq. (21) is asymptotically stable under the adaptation laws given by Eq. (22). The error signals are convergent along with time, i.e., e (t) , e˙ (t) → 0, as t → +∞. ∂ˆ hN F Sk ˙ ∂ˆ hN F Sk ˙ ∂ ˆhN F Sk ,σ ˆ kji = −Γσkji sk ,w ˆ kj = −Γwkj sk . ˆ˙ kji = −Γkji sk ∂ kji ∂σkji ∂wkj (22) ˆ N F Sk ∂ h ˆ N F Sk ˆ N F Sk ∂h ∂h Where ∂kji , ∂σkji , and ∂wkj can be detailed by N i
ˆ N F Sk ∂h ∂wkj
=
x − ˆ kji 2 exp − iniσ ˆ
ˆ N F Sk ∂h , ∂kji x − N ˆ N F Sk r N i ˆ kji 2 ∂h exp − iniσ ˆ kji ∂σkji j=1 i=1 i=1
kji
= =
ˆ N F Sk ) ˆ 2(xini − ˆ kji )(w ˆkj −h ∂ hN F Sk 2 ∂wkj σ ˆkji ˆ N F Sk (xini − ˆ kji ) ∂ h · ∂kji σ ˆkji
(23)
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Proof. Considering the following Lyapunov candidate: ¯ · s + 1 · t s (t) dt T · KI · t s (t) dt · sT · M 2 0 0 N 2 Nr n+2 2 2 i ˜ kji σ ˜kji w ˜ kj + 12 · + + ≥0 Γ Γσ Γw
VS =
1 2
k=1
j=1 i=1
kji
kji
kj
The time derivative of Lyapunov candidate is ˙ ¯ ¯ · s˙ + KI · t s (t) dt + sT ·M·s V˙ S = sT · M 2 0 N Nr n+2 i ˙ kj ˜ kji · ˜˙ kji σ ˜ kji ·σ ˜˙ kji w ˜kj ·w ˜ + Γw + + Γσ Γ k=1
j=1
i=1
kji
kji
(24)
(25)
kj
From Eq. 21, we have ¯ · s˙ + KI · t s (t) dt = −sT · V¯ · s sT · M 0 ˜ N F S + k · sgn (s) + −sT · KP · s − sT · h
(26)
ˆ˙ kji , σ ˜˙ kji = −σ ˆ˙ kji , w ˜˙ kj = −w ˆ˙ kj . Substituting Eqs. Notice that ˜˙ kji = − 19,22 and 26 into 25, and considering kk ≥ |k | + |hres | at the same time, yields V˙ S ≤ −sT · KP · s ≤ 0
(27)
From Eqs. 24 and 27, VS is a Lyapunov function. According to LaSalle’s theorem, the system is asymptotically stable and s → 0 as t → +∞. Define
p = {x (t) ∈ n : xp < ∞}. From Eqs. 24 and 27, s (t) ∈ 2 . According to Eq. 16, e (t) ∈ 2 ∩ ∞ , e˙ (t) ∈ 2 , and e (t) → 0, as t → +∞. Since V˙ S ≤ 0, ˜ kji , σ ˜kji , w ˜kj ∈ ∞ , if the Jacobian is full rank, VS ∈ ∞ , which implies that ˆ N F S ∈ ∞ . hN F S ∈ ∞ and kji , σkji , wkj ∈ ∞ , so, ˆ kji , σ ˆkji , w ˆkj ∈ ∞ and h Considering that hres ∈ ∞ , so K ∈ ∞ . Then, from Eq. 21, s˙ (t) ∈ ∞ . Since s (t) ∈ 2 and s˙ (t) ∈ ∞ , s (t) → 0 as t → +∞, which is followed by e˙ (t) → 0. End of the proof.
4
Simulation Results
The simulation is performed on a real robot composed of a 3-wheeled mobile platform and a 4-DOF modular manipulator, as shown in Fig. 1(a). In order to verify the algorithm, the robot is required to follow a spacial trajectory in Fig. 3(a), which has been planned to ensure the robot far away form singularities or joint limits. Two ball-like task-consistent obstacles with radius of 0.2m are considered, one is on the motion plane of the mobile platform and the other is on the way of the modular manipulator as shown in Fig. 3(b). The simulation time is selected as 20 seconds. Each element of h is approximated by a NFS. The gain matrices and constants are selected as follows: KP = diag {100} , KI = diag {10} , K = diag {50} , Γkji = 0.1, Γσkji =
Redundant Nonholonomic Mobile Modular Manipulators
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0.8 0.75 0.7
z
p ,m
0.65 0.6 0.55 0.5 0.45
Controlled
0.4 2
Desired
1.5
1.5 1
1 0.5
0.5 p ,m
0
0
y
−0.5 −0.5
−1
p ,m x
(a) Desired and controlled locus
(b) Obstacle avoidance
Fig. 3. Desired and control locus and obstacle avoidance
0.4
0.2
0.3
0.15
0.1
0.2
Tracking velocity errors, m/s
Tracking position errors, m
ey 0.05
ex
0
−0.05 e
z
−0.1
e•
y
0.1 e•
0
x
−0.1 •
ez −0.2
−0.15
−0.3
−0.2
−0.25
−0.4 0
2
4
6
8
10
12
14
16
18
Time, s
(a) Tracking position errors
20
0
2
4
6
8
10
12
14
16
18
20
Time, s
(b) Tracking velocity errors
Fig. 4. Tracking position and velocity errors
0.1, Γwkj = 0.1, Λ = diag {2.0} and Nr = 200. The cut-off distance is selected as dc = 0.5m, and the coefficient is determined by kφ = 1.0. The desired and the controlled locus are shown in Fig. 3(a). Two obstacles are avoided by controlling self-motions of the mobile modular manipulator in Figure 3(b). The tracking position and velocity errors are given by Fig. 4. It can be observed that the proposed algorithm is effective in both avoiding obstacles and controling the end-effector to follow a desired spacial trajectory simultaneously.
5
Conclusions
A mobile modular manipulator composed by a 3-wheeled nonholonomic mobile platform and a n-DOF onboard modular manipulator is investigated in this paper. Firstly, an integrated dynamic modeling method is presented. Secondly, a new obstacle avoidance algorithm using self-motions is proposed, which can avoid obstacles without affecting the end-effector planning task. Lastly, simulations are performed on a real mobile modular manipulator, which demonstrate that the proposed algorithm is effective. The dynamic modeling method and the obstacle
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avoidance algorithm proposed can be extended to study other kinds of mobile manipulators as well.
Acknowledgements This work was supported by the Research Committee of University of Macau under grant RG082/04-05S/LYM/FST.
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