A New Method of Executing Multiple Auxiliary ... - Semantic Scholar

Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems October 9 - 15, 2006, Beijing, China

A New Method of Executing Multiple Auxiliary Tasks by Redundant Nonholonomic Mobile Manipulators Yugang Liu and Yangmin Li 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 Email: [email protected], Email: [email protected]

of not contributing to any end-effector velocities. When the Jacobian matrix loses rank, kinematic singularity −1 will occur.  , DLS soluReplacing J † with J # = J T J J T + λ2 Im tion can be obtained, here λ is called a damping factor. In the previous research literatures, self-motions of redundant robots were mostly exploited to accomplish only one secondary task [1]–[4] but few works can be found on performing multiple secondary tasks [5], [6]. However, in practical applications, a redundant mobile manipulator may have to face up simultaneously more than one of such dangers as obstacles, joint limits, singularities and tip-over instabilities. Therefore, the study on multiple secondary tasks performing is of great requisite. Up to now, there are basically two kinds of schemes for secondary task performing: one is the gradient projection method (GPM), as shown in Eq. 3, which are normally exploited to perform only one secondary task; another scheme is called task-priority allocation method, which is very complex and may introduce algorithm singularities [4]–[6]. T  ∂w (q) q˙s = α (3) ∂q where α is a constant gain factor; w denotes a scalar objective function for the secondary task, which can be selected as:

Abstract— This paper addresses the multiple tasks performing issues for redundant nonholonomic mobile manipulators. An extended gradient projection redundancy resolution scheme is proposed which can determine the directions of self-motion to perform multiple secondary tasks. This scheme is easy to use and can avoid algorithm singularities. A general dynamic modeling method is presented in consideration of nonholonomic constraints, interactive motions and self-motions. A real-time fuzzy logic self-motion planner is devised to create desired selfmotion magnitudes and a robust adaptive neural-network controller is designed to accomplish multiple secondary tasks without affecting the primary one in the workspace. The effectiveness of the proposed algorithm is verified through simulations for a 3DOF manipulator atop a 3-wheeled mobile platform system.

I. I NTRODUCTION In the past decades, redundant robots attracted attentions from numerous researchers due to their abilities to perform complex secondary tasks in terms of hindering physical limits [1], avoiding obstacles [2], preventing overturns [3], and removing singularities [4], [5]. Several redundancy resolution schemes have been proposed, including minimum-norm solution, weighted least norm (WLN) solution [1], damped least squares (DLS) solution [4], and so on. Let q ∈ Rn , x ∈ Rm (m < n) represent respectively jointspace and task-space variables for a redundant robot, then the differential kinematics can be given by x˙ = J q˙

⎧

⎪ det (JJ T ) ⎪ ⎪

2 n ⎪  ⎪ qi −¯ qi 1 ⎪ − ⎪ ⎪ ⎨ 2n i=1 qiM −qim w (q) = min xci − xoj  ⎪ ⎪ i,j ⎪ ⎪ 3  ⎪   ⎪ 1 ⎪ ¯ 2 Ni − N ⎪ ⎩ − 3

(1)

Where J ∈ R is the Jacobian matrix. Solving Eq. 1, yields m×n

  q˙ = J † x˙ + In − J † J q˙s (2)   −1 where J † = J T · J · J T is the Moore-Penrose generalized inverse of J; q˙s is an arbitrary n-dimensional vector; In represents the identity matrix in size of n × n. † nd  In Eq.† 2, J x˙ is a minimum-norm solution; the 2 term, In − J J q˙s ∈ ℵ(J), the null space of J, is a homogeneous solution, which corresponds to the self-motions with a feature

1-4244-0259-X/06/$20.00 ©2006 IEEE

singularity removing joint limit avoidance obstacle avoidance overturn prevension

i=1

(4) where qiM and qim is the maximum and minimum limit for qi respectively; q¯i is the middle of joint range; xci and xoj represent position vectors for the ith critical point and the j th th obstacle, [16];  3 Niisthe supporting force on the i wheel,  ¯ = 3. Ni and N i=1

1

The task-priority allocation solution can be given by [6]

Let the secondary tasks to be performed be in dimension of ms1 , ms2 , · · · , msns with ms1 + ms2 + · · · + msns = n − m, where ns represents the number of secondary tasks. Assume the corresponding self-motion joint velocities be given by qs1 , qs2 , · · · , qsns ∈ Rn . Then from Eq. 2 and Eq. 6,

  †     x˙ s − Js J † x˙ + In − J † J q˙ = J† x˙ + Js In − J † J   †    Js In − J † J · In − Js In − J † J · q˙s (5) where Js ∈ ms ×n is the secondary-task Jacobian matrix; x˙ s = Js q˙ denotes the secondary-task differential kinematics. Equation 5 gives an example with only one secondary task x˙ s . This result can be extended to multiple secondary tasks, but the computational complexity will increase in an almost manner [6]. Furthermore, when the matrix  exponential  Js In − J † J loses rank, algorithm singularities will occur, which are caused by incompatibilities of the primary task x with the secondary task xs , i.e., ℵ (J) ∩ ℵ (Js ) = {0}. In addition, the secondary-task Jacobian is difficult to determine. Studying on intelligent mobile manipulators is another hot topic since they have many potential applications. Compared with such conventional manipulators as mounted on fixed bases, a mobile manipulator has much larger workspace. However, dynamic modeling and trajectory following tasks become even more complex and difficult to achieve due to the interactive motions of these two parts [7]–[9]. Furthermore, tip-over stability is another concerning issue since the probability of tip-over increases due to this combination [10], [11]. At another aspect, neural networks (NNs) have been widely used for robotic control because they are universal approximators. The combination of NNs and adaptive technique makes the controller be able to suppress errors caused by parameter uncertainties. In related literatures, real experiment for mobile manipulators using NN control was approached [12]. Adaptive NNs were designed for motion control of nonholonomic mobile robots [13]. The rest of the paper is organized as follows: an extended gradient projection method (EGPM) is proposed and a general dynamic modeling method is presented in the next section. Section III gives the fuzzy logic self-motion planner and the robust adaptive NN controller. Simulations are conducted in Section IV. Finally, Section V provides some conclusions.

q˙ = +



r  i=0

ui σi viT

vi σi−1 uTi x˙ +

i=1 m+m s1 +ms2 i=m+ms1 +1

m  i=r+1

vi viT q˙s0 +

vi viT q˙s2 + · · · +

m+m  s1

n 

i=m+1

i=n−msns +1

vi viT q˙s1 (7)

vi viT q˙sns

Where the 1 term denotes the minimum-norm joint velocities; the 2nd term represents the null space velocities due to kinematic singularities (if r < m); and the other terms are null space joint velocities used to fulfill secondary  T tasks; i the ith secondary-task joint velocity q˙si = ksi ∂w with ∂q th wi optimization function for the i secondary task; the gain matrix ksi = diag {ksi1 ksi2 · · · ksin } with ksij ≥ 0. T Define Jacobian Js1 = [vm+1 · · · vm+ms1 ] ∈ Rms1 ×n , T Js2 = [vm+ms1 +1 · · · vm+ms1 +ms2 ] ∈ Rms2 ×n , · · ·, T  msns ×n ∈ R for secondary Jsns = vn−msns +1 · · · vn tasks, then from the properties of eigenvectors we can obtain st

T T = 0m×msi , Jsi · Jsi = Imsi , J · Jsi † T Jsi · J = 0msi ×m , Jsi · J sj  T = 0msi ×msj (i = j) T · Jsi = In − J † J − Jsj Jsj Jsi

(8)

j=i

Define x˙ s1 = Js1 q˙s1 , x˙ s2 = Js2 q˙s2 , · · ·, x˙ sns = Jsns q˙sns , then Eq. 7 can be written as T T T · x˙ s1 + Js2 · x˙ s2 + · · · + Jsn · x˙ sns (9) q˙ = J † · x˙ + Js1 s   T T T T Define xE = x xs1 xs2 · · · xTsns as the extended task-space variables including both primary and secondary   T T T , then from Eq. 9 Js2 · · · Jsn tasks. Let JE† = J † Js1 s

q˙ = JE† · x˙ E

(10) JE†

Remark 1: If rank(J) = m, matrix is invertible and its inverse can be calculated as follows: −1  T T T T JE = JE† = J T Js1 Js2 · · · Jsn (11) s

II. R EDUNDANCY R ESOLUTION & DYNAMIC M ODELING To present the proposed EGPM scheme, consider the singular value decomposition of the Jacobian matrix J, [15] J = U ΣV T =

r 

Here, we will not give the proof because of page limitation. Left multiplying JE in Eq. 10, yields

(6)

x˙ E = JE · q˙



(12)

From the derivative of Eq. 12, we can obtain

Where Σ = diag {σ1 , σ2 , · · · , σm } 0m×(n−m) ∈ Rm×n with σ1 ≥ σ2 ≥ · · · ≥ σr > σr+1 = · · · = σm = 0 nonnegative square roots of the eigenvalues of matrix JJ T ; U = [u1 u2 · · · um ] ∈ Rm×m with ui (i = 1 · · · m) eigenvectors of matrix JJ T ; V = [v1 v2 · · · vn ] ∈ Rn×n with vi (i = 1 · · · n) eigenvectors of matrix J T J (arranged in the same order as the corresponding eigenvalues σi2 ); r is the rank of J; the range space and null space of J are given by  (J) = span {u1 , · · · , ur }, ℵ (J) = span {vr+1 , · · · , vn }.

¨E − JE† · J˙E · JE† · x˙ E q¨ = JE† · x

(13)

T

Define ξ = [xm ym φm φL φR ] . From Fig. 1(a), nonholonomic constraints of the mobile manipulator are given by A (ξ) · S (ξ) = 03×2 3×5

Where A ∈ 

2

5×2

and S ∈ 

(14)

; details can be found in [14].

φL

Ym

Driving Wheel

wih Xm

φm

G

dm

x1

Om ( xm , ym )

rf

w

i

QS

VS

SM

μk

NS

ZE

sij

ME

SM

BG

VB

who

θo f NN h

−1)

x Ni

θ hN

lG

h

w (q)

0

φR

(a) A 3-wheeled mobile manipulator Fig. 1.

ZE

θh 2 θh ( N

Castor Wheel

Driving Wheel

μ

θ h1

Onboard Manipulator

i

(a) The input fuzzy set (b) A 3-layered MISO MFFN

Fig. 2.

ksijM

0

(b) The ouput fuzzy set

Input and output fuzzy sets

A 3-wheeled mobile manipulator and a 3-layered MISO MFFN

T

knowledge can be incorporated and trial and error approach T can also be exploited [17]. Assume q˙ ∈ [q˙m q˙M ] , then the T output fuzzy sets can be determined ksij ∈ [ksijm ksijM ] with ksijm = 0 and ksijM selected according to the signs i (q) of ∂w∂q and the self-motion distribution strategy. Here, we present an example named weighted distribution strategy:

T

Let q = [φL φR q1 q2 · · · qN ] and x = [px py pz ] ; T  define ζ = ξ T q T , the constraint dynamics are given by   M · ζ¨ + V · ζ˙ + G = B · τ + J T · Fext + C · λ

(15)

Where the implications for M , V , G, B, C, τ , Fext , and λ can be found in [14]; N is the DOF for the onboard manipulator. From the definition of ζ, we can obtain   S 05×N ˙ζ = · q˙ (16) 0N ×2 IN

ksijM =

⎧ asi ⎪ · (q˙jM − q) ˙ , ns ⎪  ⎪ ⎪ asi ⎨ i=1

asi ⎪ · (q˙ − q˙jm ) , ns ⎪  ⎪ ⎪ asi ⎩

∂wi (q) ∂qj

>0

∂wi (q) ∂qj

0. (1) For any r ∈ RN +2 , rT · M   ˜˙ − 2V˜ · r = 0. (2) For any r ∈ RN +2 , rT · M   ˜ = Y ζ, q, ˜ ·x ˙ q¨ · φ. (3) M ¨E + V˜ · x˙ E + G ˜ , V˜ , G ˜ ∈ ∞ . (4) If rank(J) = 3, M

if if if if if

III. F UZZY L OGIC S ELF -M OTION P LANNER & ROBUST A DAPTIVE N EURAL -N ETWORK C ONTROLLER D ESIGN

wi (q) wi (q) wi (q) wi (q) wi (q)

is is is is is

ZE, V S, QS, SM, N S,

then then then then then

ksij ksij ksij ksij ksij

is is is is is

V B; BG; M E; SM ; ZE.

(19)

By applying product inference and centroid defuzzifier, the magnitudes for desired self-motions can be determined by  5 5       μwil · bksijl μwil (20) ksij =

The EGPM proposed in the last section just provides the directions of self-motions but not the magnitudes. The selection of gain factors ksi is very important. With too small ksi , secondary tasks may not be fulfilled and some dangers may occur; with too large ksi , physical limits of the robot may be passed over. In this paper, the FLSMP is designed to determine the magnitudes for the self-motion joint velocities. In these fuzzy logic systems, the inputs are the objective functions wi (q), while the outputs are the gain factors ksij . The discourse for input fuzzy variable wi can be determined according to the concrete secondary tasks, in which human

l=1

l=1

Where μwil is the input membership function with respect to the lth rule; bksijl are corresponding centroids of the consequent fuzzy sets for the lth rule.  T i , the desired selfSubstituting ksij into q˙si = ksij ∂w ∂q motion joint velocities can be obtained; from Eq. 9, the desired ¨sid value for x˙ si , i.e., x˙ sid can be determined. Then, xsid and x can be calculated via the integral and differential operations.

3

Assume xd , x˙ d and x ¨d be the desired trajectory, velocity and acceleration in task space. Let xEd = [xTd xTs1d · · · xTsns d ]T , then the error system can be defined as follows e (t) = xE (t) − xEd (t) x˙ r (t) = x˙ Ed (t) − Λ · e (t) r (t) = x˙ E (t) − x˙ r (t)

q

ζ

ζ

Fuzzy Logic SelfMotion Planner

−

xEd

+

ˆNNk ∂h ∂θkhj

e

+

Kε

Σ

xr − +

r

Σ

− −

KP

Σ

−

+

Σ

−

JT

KI

xE

Fig. 3.

Fext

A robust adaptive neural-network controller

The RANNC is given by Eq. 26, and the control system is shown in Fig. 3. =

τ

  ˆ N N − KI t r (t) dt JET h 0  −KP · r − K · sgn (r) − J T · Fext



S¯T B

−1

(26)

ˆ N N forms the adaptive NN term; KP , KI ∈ Where h (N +2)×(N +2) are proportionaland integral gain matrices of   the PID controller; K = diag k1 k2 · · · kn is the gain matrix of the robust term with kk ≥ |hres | + | k |. Substituting Eq. 26 into 22, and considering Eq. 17, yields:

˜ r+ M ˙ V˜ r+KP ·r+KI



t

0

˜ N N + = 0 r (t) dt+K sgn (r)+h (27)

Theorem 1: If KP > 0, = KI > 0, the closed-loop system in Eq. 27 is asymptotically stable under the adaptation laws given by Eq. 28. The error signals are convergent with time, i.e., e (t) , e˙ (t) → 0, as t → +∞. KIT

(24)

ˆNNk ˆNNk ˙ ∂h h , θˆkhj = −Γθkhj rk ∂∂θ , w ˆ˙ kihji = −Γwkihji rk ∂w kihji khj ˆNNk ˆNNk ˙ ∂h ∂h ˙ ˆ w ˆ khoj = −Γwkhoj rk ∂wkhoj , θko = −Γθko rk ∂θko . (28) Where Γwkihji , Γwkhoj , Γθkhj , and Γθko are positive constants. The terms with partial differentiation can be derived from Eq. 23, details will not be listed here.

· θ˜ko

 θ˜khj + hres

Λ

(22)

Where hN N k = fN N (xin , wkih , wkho , θkh , θko ); k is the approximated error; k = 1, 2, · · · , N + 2. Let w ˆkih , w ˆkho , θˆkh and θˆko be estimates of wkih , wkho, ˆ N N k = fN N xin , w ˆkih , w ˆkho , θˆkh , θˆko , θkh and θko , define h ˆ N N k is then the Taylor series expansions of hN N k around h

w ˜khoj +

q xE

A mobile manipualtor

xd

Where ϕ (◦) is the activation function named hyperbolic tangent function; Ni and Nh represent neuron numbers; x are the inputs; wih , who , θh and θo denote weights and thresholds accordingly; the subscript “i, h and o” represents the input, hidden and output layer respectively. T  ¨TEd , from Remark 2 Define xin = ζ T q˙T xTEd x˙ TEd x ˜ ·x ˜ and Eq. 21, all the elements h (xin ) = M ¨r + V˜ · x˙ r + G are bounded as long as J keeps full rank. And they can be approximated by MISO MFFNs,

j=1 i=1 N h  ∂ hˆ N N k + ∂wkhoj j=1

Σ

−

[ S T B ]−1 J ET

xs1d

ζ

xE

Adaptive Law

+

xd

i=1

ˆNNk ∂h ∂θko

∂hˆNN ∂ θ h MFFN ∂hˆ ∂ θ NN o

xE

xsns d

It is verified that multilayer feed-forward network (MFFN) with as few as one hidden layer using arbitrary squashing activation functions and linear or polynomial integration functions can approximate virtually any (Borel-measurable) functions to any desired degree of accuracy, provided sufficiently many hidden units are available [17]. To ensure rapid convergence, 3-layered multiple-input single-output (MISO) MFFNs are adopted, as shown in Fig. 1(b). The output of this MFFN is:  N   Nh i   xi · wihji + θhj · whoj + θo (23) ϕ fN N =

Nh Ni ˜ N N k =   ∂ hˆ N N k · w h ˜kihji + ∂wkihji

xNi

∂hˆNN ∂ who

xEd

xsns d

hk (xin ) = hN N k (xin ) + k (xin )

y NN

θˆh

xEd

˜ · r˙ (t) + V˜ · r (t) + M ˜ ·x ˜ = τ˜ M ¨r + V˜ · x˙ r + G

∂hˆNN ∂ wih

x1

θˆo

xsns d xd xs1d

Where r (t) is the tracking error measure; Λ > 0 is a constant matrix in size of N + 2 by N + 2. Substituting Eq. 21 into Eq. 17, yields:

j=1

wˆ ih wˆ ho

xs1d

(21)

hˆNN

xin

q xEd xEd xEd

(25)

Proof: Considering the nonnegative Lyapunov candidate:

ˆkihji , w ˜khoj = wkhoj − w ˆkhoj , Where w ˜kihji = wkihji − w ˜ N N k = hN N k − h ˆNNk; θ˜khj = θkhj − θˆkhj , θ˜ko = θko − θˆko , h N Ni  2    2   2  h   2  hres = O w ˜khoj +O θ˜khj + O w ; ˜kihji +O θ˜ko   2  j=12   2   2 i=1 ˜khoj , O θ˜khj and O θ˜ko are higherhere O w ˜kihji , O w order terms.

VS =

˜ r rT M 2

+ 21

4

 t +

0

T

r(t)dt

N +2 N h

Ni 

k=1 j=1

i=1

KI

 t

2 2 w ˜kihji Γwkihji

+

0



r(t)dt

2 w ˜khoj Γwkhoj

+ +

1 2

N +2 ˜2 θko Γθko k=1 !

2 θ˜khj Γθkhj

≥0 (29)

The time derivative of Lyapunov candidate is V˙ S = rT + 12

   ˜ r˙ + KI t r (t) dt + M 0

N +2 N h

Ni 

k=1 j=1

i=1

˙ kihji ˜ w ˜kihji w Γwkihji

+

˙ r ˜ rT M 2

+

˙ khoj ˜ w ˜khoj w Γwkhoj

convergent properties of the MBC can be verified, which will not be detailed here; the damping factor is selected as, [4]: " 0 σmin > δ

2 2 (33) λ = σmin 2 λmax σmin < δ 1 − δ2

N +2 ˜ ˜ ˙ θko θ ko Γθko k=1 ! ˙ θ˜khj θ˜ + khj Γθkhj

1 2

Where σmin is the minimum singular value of J; δ = 0.05 and λmax = 0.07 are constants selected to provide best task performance [5]. The simulation time interval is selected as 20 seconds with the sampling frequency 1kHz. All joint angles and velocities are initialized to be zeros. These two schemes are exploited to command the end-effector to follow a sine link spacial trajectory, as shown in Fig. 4(a). The simulation results are presented in two parts: the desired and controlled locus for these two schemes are shown in Fig. 4(a) and Fig. 4(b) respectively; Figure 4(c) and Fig. 4(d) give the position errors; velocity errors for these two schemes are presented by Fig. 4(e) and Fig. 4(f) accordingly; the optimization functions for the two secondary tasks are given by Fig. 5(a)–5(d) respectively; at last Fig 5(e)–5(f) show the time-variant control torques for these two schemes. From these figures, we can conclude that the proposed algorithm is effective for multiple secondary tasks performing. Compared with the scheme using conventional GPM, DLS resolution and model-based control, the proposed algorithm behaves better.

(30) ˙ ˙ ˙ ˙ ˆ˙ khoj , θ˜khj = −θˆkhj , θ˜ko = −θˆko , Notice that w ˜˙ khoj = −w ˙ ˙ ˆ kihji , substituting Eq. 25 into Eq. 27, then w ˜ kihji = −w substituting the result together with Eq. 28 into Eq. 30, and considering Remark 2 at the same time, yields V˙ S ≤ −rT · KP · r ≤ 0

(31)

Therefore VS is a Lyapunov function, iff r = 0, VS and V˙ S equal to zero. According to LaSalle’s theorem, the system is asymptotically stable and r → 0 as t → +∞. Define p = {x (t) ∈ n : xp < ∞} the p − norm space. From Eqs. 29 and Eq. 31, r (t) ∈ 2 . According to Eq. 21, e (t) ∈ 2 ∩ ∞ , e˙ (t) ∈ 2 , and e (t) → 0, as t → +∞. It is obvious that the higher-order terms in Eq. 25 are bounded, so K ∈ ∞ . Then, from Eq. 27, r˙ (t) ∈ ∞ . Since r (t) ∈ 2 and r˙ (t) ∈ ∞ , r (t) → 0 as t → +∞, which is followed by e˙ (t) → 0. End of proof. IV. S IMULATION R ESULTS

Remark 3: In this simulation, the task-space information x and x˙ are calculated respectively by forward kinematics and differential kinematics. Since the sampling time interval Δt = 0.001s is so small compared with the total time interval of 20s, the simulation results, especially those for the DLS based MBC algorithm, give one an impression that the results are not continuous. However one need not worry about that matter because they are all continuous and smooth when locally enlarged. Comparing Fig. 5(c) with Fig. 5(d), we can see that the DLS based MBC algorithm performs even better than the proposed algorithm for the secondary task of tipover avoidance at the time interval 8s–12s; that is because the former exploiting 2 degree of redundancy for tip-over avoidance but the proposed algorithm has only one degree of redundancy for that secondary task.

The mobile manipulator adopted in this simulation is composed of a 3-wheeled nonholonomic mobile platform and a 3-degree of freedom (DOF) onboard manipulator, in the form of Fig. 1(a). Since only the end-effector positions are concerned, the degree of redundancy is 2. Then, besides endeffector path following, there are at most 2 secondary tasks can be performed, singularities removing and tip-over prevention here. To verify the effectiveness of the proposed algorithm, simulations are carried out for two kinds of schemes: (1) The proposed algorithm, with ms1 = ms2 = 1, w1 and w2 are selected according to Eq. 4; for FLSMP, as1 = as2 = 0.5; regarding to the RANNC, the gain matrices and constants are selected as: Nh = 200, KP = 25 I5 , KI = 10 I5 , K = 10 I5 , Λ = 2 I5 , Γwkihji = Γwkhoj = Γθkhj = Γθko = 0.01. (2) A DLS based model-based controller (MBC) with GPM redundancy resolution, as shown in Eq. 32; the DLS solution with numerical filtering is introduced, as given by Eq. 33, which was verified to give a good compromise between task precisions and singularities avoidance [4]; conventional GPM redundancy resolution scheme is adopted as shown in Eq. 3.   −1  ¯ J# x M ¨d − J˙ q˙dls − KD e˙ x τdls = S¯ B   ¯ − J T Fext −KP ex + V¯ q˙dls + G

V. C ONCLUSION A practical algorithm is presented for multiple secondary tasks performing of redundant mobile manipulators. An extended gradient projection method is proposed, which can be exploited to accomplish multiple secondary tasks without introducing algorithm singularities. Magnitudes of the selfmotion are created by fuzzy logic self-motion planner but not just selected as constants. A robust adaptive neural-network controller is developed in task space, which can command the end-effector to follow desired trajectories and control the selfmotions to perform secondary tasks at the same time. The simulations are carried out on a 5-DOF mobile manipulator. However, the proposed algorithm can easily be extended to some other kinds of redundant robots as well.

(32)

  ∂w2 (q) T ˙ I5 − J # J Where q˙dls = J # x+α , ex = x−xd , ∂q  T −1 # T 2 JJ +λ I ; α = 0.1 is a constant gain factor; J =J KD = 10 I3 and KP = 25 I3 are constant gain matrices; the 5

0.5

0.5

controlled

0.4 5

desired

5

−5

5

2

0 px, m

y

0

p ,m y

−2

(a) Locus for the proposed algorithm

−5

0 px, m

(b) Locus for the DLS based MBC

0.2

ey

ex

0.1

ez

e

−0.1

y

0 −0.1 −0.2 ex

−0.3

−0.2

−0.4 −0.3

0

5

10 Time, s

15

−0.5

20

0

5

10 Time, s

15

20

(c) Position errors: proposed algorithm (d) Position errors: DLS based MBC 0.5

e•x •

ez −0.5

5

10 Time, s

15

20

0 −0.5 e•y

−1

−2

1

4

2

0

Fig. 4.

10 Time, s

15

20

0

−50

−100 w (q) 2

−150

0

5

10 Time, s

15

20

0

5

10 Time, s

15

(f) Velocity errors: DLS based MBC

Simulation results : part I

10 8 6 4 2 0

w (q) 1

0

5

10 Time, s

15

20

(b) w1 (q) for the DLS based MBC 0

−50

−100 w (q) 2

−150

−200

0

5

10 Time, s

15

20

(d) w2 (q) for the DLS based MBC

20 τ

2

0 τ

−5

τ

τ

3

L

1

τ

−10

20

×10

12

40

5

−15

(e) Velocity errors: proposed algorithm

5

10

−1.5 0

w (q)

15

e•x

0.5

Control torques, N.m

Velocity errors, m/s

Velocity errors, m/s

e•z

1

y

0

6

(c) w2 (q) for the proposed algorithm

1.5 e•

−1

Optimization function for tip−over avoidance

0.2 Position errors, m

ez

0

8

(a) w1 (q) for the proposed algorithm

0.3

0.1 Position errors, m

controlled

0.4 4

desired 0 p ,m −5

0.6

10

Optimization function for singularity removing

z

0.6

−3

Optimization funciton for tip−over avoidance

0.7

−3

x 10

12

Control torques, N.m

0.7

Optimization function for singularity removing

0.8

p ,m

z

p ,m

0.8

τ τ

−20

1

τ

3

2

τ

R

τ

−40

R

0

0

5

10 Time, s

15

20

−60

L

0

5

10 Time, s

15

20

(e) Control torques: proposed algorithm (f) Control torques: DLS based MBC Fig. 5.

Simulation results : part II

VI. ACKNOWLEDGEMENT [8] Q. Yu and I-M. Chen, “A general approach to the dynamics of nonholonomic mobile manipulator systems”, ASME Trans. Dyn. Syst., Measur., Contr., vol. 124, no. 4, pp. 512–521, 2002. [9] J. Tan, N. Xi and Y. Wang, “Integrated task planning and control for mobile manipulators”, Int. J. Rob. Res., vol. 22, no. 5, pp. 337–354, May, 2003. [10] A. Ghasempoor and N. Sepehri, “A measure of machine stability for moving base manipulators”, in Proc. of IEEE Int. Conf. on Rob. and Autom., pp. 2249-2254, 1995. [11] Y. Li, and Y. Liu, “Online fuzzy logic control for tipover avoidance of autonomous redundant mobile manipulators”, Int. J. Vehicle Autonomous Systems, vol. 4, no. 1, pp. 24–43, 2006. [12] L. Sheng and A. Goldenberg, “Neural-network control of mobile manipulators”, IEEE Trans. Neural Netw., vol. 12, no. 5, pp. 1121–1133, Sep., 2001. [13] R. Fierro and F. L. Lewis, “Control of a nonholonomic mobile robot using neural networks”, IEEE Trans. Neural Netw., vol. 9, no. 4, pp.589– 600, Jul., 1998. [14] Y. Li, and Y. Liu, “Fuzzy Logic Self-motion Planning and Robust Adaptive Control for Tip-over Avoidance of Redundant Mobile Modular Manipulators”, in Proc. IEEE/ASME Int. Conf. on Adv. Intelligent Mechatron., California, USA, pp. 1281–1286, Jul., 2005. [15] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1991. [16] C. C. de Wit, B. Siciliano and B. Georges, Theory of Robot Control, Springer-Verlag London Ltd., 1996. [17] C. T. Lin and C. S. G. Lee, Neural Fuzzy Systems: A Neuro-Fuzzy Synergism to Intelligent Systems, Prentice-Hall Int., Inc., 1996.

The authors appreciate the fund support from the research committee of University of Macau under grant no.: RG082/0405S/LYM/FST. R EFERENCES [1] T. F. Chan and R. V. Dubey, “A weighted least-norm solution based scheme for avoiding joint limits for redundant joint manipulators,” IEEE Trans. Rob. Autom., vol. 11, no. 2, pp. 286–292, Apr. 1995. [2] O. Brock, O. Khatib and S. Viji, “Task-consistent obstacle avoidance and motion behavior for mobile manipulation”, in Proc. of IEEE Int. Conf. on Rob. and Autom., Washington, DC, pp. 388–393, May, 2002. [3] Y. Li and Y. Liu, “A new task-consistent overturn prevention algorithm for redundant mobile modular manipulators”, in Proc. of IEEE/RSJ Int. Conf. on Intelligent Rob. and Syst., Canada, pp. 1563–1568, Aug., 2005. [4] S. Chiaverini, “Singularity-robust task-priority redundancy resolution for real-time kinematic control of robot manipulators,” IEEE Trans. Rob. Autom., vol. 13, no. 3, pp. 398-410, Jun. 1997. [5] J. Kim, G. Marani, W. K. Chung, J. Yuh and S. R. Oh, “Dynamic task priority approach to avoid kinematic singularity for autonomous manipulation”, in Proc. IEEE/RSJ Int. Conf. Intelligent Rob. and Syst., Switzerland, pp. 1942–1947, Oct., 2002. [6] D. N. Nenchev and Z. M. Sotirov, “Dynamic task-priority allocation for kinematically redundant robotic mechanisms”, in Proc. IEEE/RSJ/GI Int. Conf. Intelligent Rob. and Syst., pp. 518–524, Sep., 1994. [7] Y. Yamamoto and X. P. Yun, “Effect of the dynamic interaction on coordinated control of mobile manipulators”, IEEE Trans. Rob. Autom., vol. 12, no. 5, pp. 816–824, Oct. 1996.

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