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International Journal of Advanced Robotic Systems

Mobile Parallel Manipulators, Modelling and Data-Driven Motion Planning

Regular Paper

Amar Khoukhi1-2-* and Mutaz Hamdan1

1 Department of Systems Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

2 Visiting Professor, DeVry College of New York, New York, NY

* Corresponding author E-mail: amar@kfupm.edu.sa Received 17 May 2012; Accepted 8 May 2013 DOI: 10.5772/56628

© 2013 Khoukhi and Hamdan; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract This paper provides a kinematic and dynamic analysis of mobile parallel manipulators (MPM). The study is conducted on a composed multi-degree of freedom (DOF) parallel robot carried by a wheeled mobile platform. Both positional and differential kinematics problems for the hybrid structure are solved, and the redundancy problem is solved using joint limit secondary criterion-based generalized-pseudo-inverse. A minimum time trajectory parameterization is obtained via cycloidal profile to initialize multi-objective trajectory planning of the MPM. Considered objectives include time energy minimization redundancy resolution and singularity avoidance. Simulation results illustrating the effectiveness of the proposed approach are presented and discussed.

Keywords Mobile Parallel Manipulator (MPM), Kinematic Redundancy Resolution, Off-Line Trajectory Planning

1. Introduction 1.1 Motivation

A mobile parallel manipulator consists of a mobile platform carrying a parallel manipulator. Mobile robots

have wide applications in such areas as automatic material handling in warehouses, transportation and health care in hospitals, and exploration in hazardous environments. Because of their high accuracy, velocity, stiffness, and payload capacity the progress of parallel robots (PRs) is accelerated since PRs outperform their serial counterparts [1]. However, the main drawback of PRs is their limited workspace, which restricts their applications [2]. Recently, many researchers have proposed parallel mobile robot design mechanisms [3-7]. Since an MPM possesses the advantages of both a mobile robot and a parallel robot, it is a potential competitor in extensive applications where high accuracy operation, a high rigidity and payload capacity are required, such as in autonomous guidance vehicles and personal robots, and space and underwater robotics.

1.2 Related Works

A literature survey on mobile parallel robots reveals that parallel mobile robots remain a hot area of research. Yamawaki et al. proposed a self-reconfigurable parallel robot, which can be configured to 4R and 5R closed kinematic chains [4]. They proposed a parallel mechanism mobile robot mounted on a crawler

mechanism. The combined mobile robot can gain some useful functionalities besides locomotion from the advantage of its parallel mechanism. For example it may be able to traverse a hump by controlling its centre of gravity, carrying an object by making use of its shape. The same study analysed the motions of these functionalities and verified them experimentally with real robots. Graf and Dillmann [5] proposed the use of a Stewart platform mounted on a mobile platform to compensate the unwanted accelerations. The necessary movement of the platform is calculated by a so-called washout filter. This combination can be applied either in the transport of liquids in open boxes or in medical transport, where patients must not be affected by any acceleration. Decker et al. implemented and compared several different approaches for the motion planning of the Gough-Stewart Platform mounted on a mobile robot[6].They aimed to enhance the capabilities of transport vehicles to enable them to carry delicate objects of various shapes and sizes without requiring extensive packaging to protect them. Li et al. proposed a novel design and modelling of mobile parallel manipulators (MPM) [15]. An MPM composed of a three-wheeled nonholonomic mobile carrier and a 3-RRPaR translational parallel platform is designed and investigated in detail. The position kinematics solutions are derived and the Jacobian matrix relating output velocities to the actuated joint rates is generated. In [6] and [7] a combination of a mobile robot and a Stewart platform is proposed for active acceleration compensation to enable the smooth transport of objects. In [8], a mobile parallel manipulator (MPM) is proposed using the combination of a wheeled mobile platform and a parallel manipulator, which provides extra mobility to the robotic system,enlarging the effective workspace of the parallel manipulator while retaining its advantages. MPMskeepthe advantages of both the mobile and the parallel robot. The MPM is a potential competitor in applications where high-accuracy operation and high rigidity and payload capacity are required, for example in autonomous guidance vehicles or personal robots. Nevertheless, the integration of a parallel manipulator and a mobile robot induces a large number of challenging difficulties, since in most cases the mobile robot is subjected to nonholonomic constraints and the parallel robot introduces many complex kinematic constraints. These difficulties include how to establish the dynamic model of the hybrid system in a systematic way, which involves redundancy resolution and singularity avoidance and often performance indicators of the controller motion. The optimal control and multi-objective trajectory planning of robotic systems is a dynamic research area [9-19]. The present paper extends the application of optimal control and multi-objective decision making to constrained time-energy trajectory planning of mobile parallel manipulators.

The remainder of this paper is organized as follows. Section 2explains in detail the kinematic modelling. In Section 3 the dynamic modelling of the hybrid system is performedusing the Lagrange method. The redundancy resolution is achieved through joint limits and singularity avoidance. A model multi-objective motion is then carried out for the MPM in Section 4. In Section 5, the ANFIS-based planning in developed. Simulations illustrating the effectiveness of the proposal are presented in Section 6. Finally, Section 7 concludes the paper.

2. Kinematic Modelling and Analysis

2.1 Architecture Description

Research into mobile parallel manipulators is a relatively new research domain, and substantial literature has recently been published [18-23]. In this paper we use the architecture presented in [2], which consists of a three-wheeled nonholonomic mobile robot and a modified 4-DOF MPM version of a DELTA parallel robot (Fig.1). Figure 2 presents the schematic diagram of the designed MPR. R and Pa stand for the revolute and parallelogram joints, respectively [2].

Moving Platform

Figure 1. The mobile parallel manipulator [2]

2.2 Position Kinematics Analysis

From Figure 2, we can obtain the following:

= _(pL + 4-) •ym^^ + Pr) (1)

, 0m = § 0r~ 6l)

Where c stands for cosine, s stands for sine, r is the radius of each driving wheel, and Ql and Qr denote respectively the rotating angles of the left and right driving wheels. Let ^ = [xm ym <pm0L 0r]T be the general coordinates of the mobile platform. The forward kinematics problem is very complex for a parallel robot, although the inverse kinematics problem is extremely straightforward in general [10].

Assuming that (vm) is the linear and (wm) the angular velocity of the mobile platform and the actuated inputs of

the actuators (di, i =1, 2, 3), the position (x, y, z) and orientation of the mobile platform are solved using the forward kinematics.

Assume that the wheels of mobile platform have no slip in the forward, reverse, or sideways directions. Let At ^ 0; the velocities during this time interval can be considered as a constant (see Fig. 2b).

Let P = [x y zf and SP = [xP Jv zp]T denote the vectors of point P in the fixed frame O and the moving frame B, respectively.

In frame B, let Bet = ~BAU Bbt = AB and Bct = PC¡. Referring to (15) we can then obtain:

Xp + (yp + u — e — bc91)2 + (zp — bs8{)2 = a2, (xp — u + e + bc02)2 +yp + (zp — bs02)2 = a2, xp + (yp —u + e + bc03)2 + (zp — bs03)2 = a2,

Using Maple (the computer algebra software system) and making some substitutions, solving (2)-(4) leads to solutions for the forward kinematics of the parallel robot:

(a) Orthographic view

(b) Top view Figure 2. Schematic representation of the MPR [2]

= _ni = _no f = m2n1 _ mt

= = >Jl= '

^ n2 o n2 m3n2 m3

m2no mo 2 2

fo=---, h2 = 1 + e{+fl2,

Jo m3n2 m3 2 1 Jl'

= 2[e1(eo + u-e-bcdi) + fifo-bs01], ho

= (eo+u_e_ bc61)2 + f2 _ a2 + b2s2d1, m3 = u _ e _ bcd2,m2 = u _ e-bcd1,m1 = b(sd1 _ sdi), mo = b(u _ e)(cd2 _ cd1),n2 = 2(u _e)_ b{c61 + c83),n1 = b(sd1 _ sd3),no = b{u_e){c63_c61),

The position of the moving platform can be derived by:

p = b + R Bp (6)

b = 0B = [xm_lbc$m ym_lbs$m h]T, (7)

S$m C(pm 0 0

is the rotation matrix of the moving frame B with respect to the fixed frame O.

2.2 Differential Kinematics Analysis

Let x = [x yz $]T be the vector for the output velocities of the moving platform, and q = [¿i 0r01 02 03]T represent the vector of the input joint rates. Differentiating (6) in terms of time leads to

p = b + RBp + RBp

Additionally, let qp = [¿1 ¿2 ¿3]Tdenote the vector of actuated joint rates for the parallel robot. Differentiating both sides of (2)-(4) with respect to time and rewriting them into a matrix form yields

ABp =Bq„

where the 3*3 forward and inverse Jacobean matrices A and B of the parallel robot can be respectively expressed as

The detailed expressions of aij and bijappear in the appendix.

a11 a12 a13' \bu 0 0

A= a21 a22 a23 ,B = 0 b22 0

a31 a32 a33. 0 0 b33

= f I f = I = ^h^-4h2h0 (5)

Xp = J1zp + J0'yp = e1zp ' e0'zp = 2h (5)

When the parallel robot is away from the singularity, from (10) we can obtain

p = JPqP

r5 = (xp - u + e + èc02)2 + yp2 + (zp - ès02)2 - a2 (26)

where Jp = A 1B is the Jacobian matrix of a 3-RRPaR parallel robot.

Substituting (7), (8) and (12) into (9), and considering (1), we obtain:

x = Jq (13)

The matrix J appears in the appendix.

Differentiating (13) with respect to time yields

x = Jq + Jq (14)

In addition, solving (13) leads to

q = j+x + (l5x5-j+j)qs (15)

where J+ = JT(JJT)_1 is the generalized pseudo inverse of J and qs £ "K5x1 is an arbitrary vector which can be chosen to achieve a secondary task.

2.3 Redundancy Resolution through Joint Limits Avoidance

To include a secondary task criterion by a performance index g(q), qs in (15) is chosen to be qs = ±fcVg(q), where k is a positive real number and Vg(q) the gradient of g(q), with the positive sign indicating that the criterion is to be maximized and a negative sign indicating minimization. To avoid joint limits we chose qs as follows:

Where:

qsi = (qmax - q)w(q - qmin)

W 2 (qmax qmin)

The related criterion to avoid the singularity is to maximize the manipulability; we choose qs as follows:

qs2 = Vdet (J J')WS (21)

where Wsis weight vector with appropriate dimension.

Now, the formula of the augmented function to avoid singularity and joint limits is as follows:

qs = qsi + qS2 (22)

3. Constraints and Dynamics Modelling 3.1 Constraint Equations:

The mobile robot cannot move in the lateral direction, and the three constraints for the mobile platform can be represented by (2). From (3) and (4), we can derive another three constraint equations for the MPM:

r6 = x2 + (yp - u + e + bc03)2 + (zp - bs03)2 - a2, 3.2 Dynamic Equations:

All the dynamic equations are taken from [8], and are re-derived here for quick reference. The Lagrange function for the MPM becomes

^ = ^m + 7p fp

The constrained dynamics for the entire system of the MPM can be determined by

d dt \

:(t)-f = <?; + ^f, (^■ = 1'2.....n) (28)

where Z = xy Z]T is the generalized

coordinates and Q = [0 0 0Ti Tr TiT2 t3 00 0]t are the generalized forces under the assumption of no external forces/torques exerted; (i = 1, 2,..., 6) are Lagrange multipliers, which can be calculated from the first set of linear equations of (27) for j = 1, 2, 3, 9, 10, and 11.

Now substitute xp and yp in (5) and solve for Zp; solving eq. 6 and eq. 7 leads to solutions for the forward kinematics of the parallel robot:

Solving the augmented Lagrange equation gives:

<?i = *4glj(0 + ^g2j(Z) + ^6g3j(0 + g4j(?)'

where i=1, 2, 3,..., 11

H(Z) = diag([Al7 A2, A4,A7 ,A10 ]) £ %5x5is the symmetrical and positive definite inertial matrix.

A1 = A2 = 9/20000, A4 = A7 = A10 = 11/750

V(Z, Z) £ ^5x5is the centripetal and Coriolis forces matrix, here equal to zero.

G(Z) = [0; 0; As; A8; A11] £ ft5x1represents the vector of gravity forces,

r4 = xp + (yp + u - e - èc01)2 + (zp - ès01)2 - a2,

A5 = 0.8829 cG,, A8 = 0.8829 cG2, Au= 0.8829 cG3

1 = 12... 16] G "K6x1denotes the vector for Lagrange multipliers 1. These multipliers' derivation and expressions are very lengthy and are not presented here.

Then, the actuated torques r= [ Tr Tir2r3]Tcan be solved from the second set of equations of (28) for j = 4, 5, 6, 7, and 8, which can be written into a matrix form:

u = H(Z)v + V(Z,Z)x2 + G(Z)

H(Z)q + V(Z,Z)q + G(Z) = r + C(Z)A

whereH(Z) G "K5x5 is the symmetrical and positive definite inertial matrix, F(Z, Z) G "W5x5 is the centripetal and Coriolis forces matrix,G(Z) G "K5x1 represents the vector of gravity forces, X = [A1 X2... X6] G *K6x1 denotes the vector for Lagrange multipliers, and C(Z) G "K5x1 is the parameter matrix for A.

Differentiating (17) with respect to time yields

x = Jq + Jq (30)

Let Jn G "K5x1 be the normalized base of n(J), which is the null space of J; then, we have

JJ„ = 04xi, JTJ„ = 1, (31)

JTJ+ = 0ix4, J„JT = I5x5 - J+J (32)

When x„ = jTqs and taking (30), (31) and (32) into consideration, we can obtain:

q = J+x + J„x„ q = J+x — J+JJ+x + J„x„ — j+Jj„x„ (33)

Defining xE = [xT x„T]T, and J+ = [J+ J„] then substituting (33) into (29) allows the derivation of the dynamic equations described in Cartesian space:

H(Z)xE+H(Z,Z)xE + H(Z) = T

H(Z) = I+TH(Z)I+, v(z,z) = I+T [v(z, Z) - H(Z)J+Tj] J+, H(Z) = I+T[G(Z) - C(Z)X], and T = j+tt.

4. Multi-Objective Motion Planning

4.1 Constrained Linear-Decoupled Form

Here,we use the same approach used by Khoukhi et al. [16]. The major computational difficulty in this system cannot be solved with the original non-linear formulation. Instead, it is solved using a linear-decoupled formulation.

Theorem:

Provided that the inertia matrix is invertible, then the control law in the Cartesian space is defined as:

This leads the robot to have a linear and decoupled behaviour with the dynamic equation:

x2 = v

where v is an auxiliary input.

This follows simply by substituting the proposed control law (39) into the dynamic model (34), obtaining:

H(0x2 = H(0w (40)

Since H(^) is invertible, it follows that x2 = v

This gives the robot the decoupled and linear behaviour described by the following linear dynamic equation, written in discrete form as:

xk+i = Fdkxk + B(ftfc)(vfc) = /d.(xfe,wfe,hk) (41)

^5x5 ^fc^5x5 .05x5 ^5x5

Notice that the non-linearity is transferred to the objective function. One problem with this formula is the lack of accuracy of the Euler's method. In order to improve the accuracy, and because the MPM structure contains highly nonlinear equations as shown in the previous chapters, we use the Adams-Bashforth Formula:

y1+1 =yi +|(3fi — fi_1)+12h3'fi (42)

Applying the Adams-Bashforth Formula (42) gives:

X1k+1 = X1k + 1.5hkX2k + 0.5hkX2k_1 + (43)

X2k+1 = X2k + 1.5hkV2k + 0.5hk^2k_1 +!7^2k (44)

Since it is difficult to obtain the derivative of vk, to improve the accuracy, the following formulas from numerical differentiation methods are used:

y1 = (y2 — yD/h! (45)

y2 =(y3— y1)/2h2 (46)

Yk = ( yk+2 + 8yk+1 — 8yk_1 + yk_2)/12hk (47)

yN_1 = (yN — yN_2)/2h2 (48)

yn = (yn — yN_1)/h2 (49)

Now, the decoupled formulation transforms the discrete optimal control problem into optimal sequences of sampling periods and acceleration inputs h0,h2, .,hN_1, v0,v2, .,vN_1, allowing the robot to move from an initial

state xo = xs to a final state xn = xt, while minimizing the cost:

E° = Minvev {sN=1 [[H(Ov + V(Ç,Ç)x2 + G(Ç)]TU[H(Ç)v V(Ç, $)x2 + H«)] + t + xTfcQx2k + 8w(xik)]hk}

The above-mentioned constraints remain the same, except actuator torques, which become:

-min < H«)v + % $)x2 + G(0 < Tmax (51)

Henceforth, inequality constraints g3 and g4 are rewritten as:

g?(xk,vk) = -min " [H(Ov + V(C ?)x2 + G(0] < 0 (52)

g?(xk,vk) = [H(Ov +V(?,?)x2 +G®]-Tmax < 0 (53)

similarly to the non-decoupled case, the decoupled problem might be written in the following form:

subject to = /^O^T^hJ, k = 0.....N - 1

flr?(*k,WfcA) <0, je {1,2.....J}

sD(xk) = 0, i e {1,2.....1}, k = 0.....N

4.2 Augmented Lagrangian Formulation

We used the same approach as [24]. Now, the augmented Lagrangian is associated to the decoupled formulation (P) as follows:

L°(x,v,h,A,p,a) =

^{[[G(Z)v + H(Z,Z)x2 + G(Z)]TU[G(Z)v + V(Z Z)x2 + G(Z)] k=0

+ xTkQx2k N-1

+ X {XT+1 (xk+1 - fdk(xk' Tk' hk))} k=0

N-1 rL-1 2

k=0 L 1=1 i=1

+ Z|=1*^s(pk'gD(xk'Tk'hk)) + S2=1hN^^s(^N.s1D1(xk))

where the function fdk(xk,Tk,hk)is defined by eq. (41) at time k, and N is the total sampling number. The other parameters appearing in (49) are defined above.

The first-order Karush-Kuhn-Tucker optimality necessary conditions require that for xk,vk,hk,k = 0, ...,N to be the solution to the problem (P), there must exist some

positive Lagrange multipliers (A.k, pk), unrestricted sign multipliers ffk, and finite positive penalty coefficients ^ = ^s), such that equation (40) is satisfied for the decoupled formulation.

The co-states A.kare determined by backward integration of the adjunct state equation, yielding:

V1 = -2hk a[V(0v+H(aK+G(0] u[H(Z)v + H(z, Z)x2 + G(Z)] - 2Qx2khk - SVXik«(x1k) - F^k -hk[ZL==11S2=1VXk^,s(ak,sD1(xk))]-hk [£|=1Vxk^g (pk,g]D(xk,Vk,hk))] (56)

The gradient of the Lagrangian with respect to sampling period variables is:

VhkLD = [[H(Z)v + H(Z, Z)x2 + G(Z)]TU[H(Z)v + H(Z, Z)x2 +

G(Z)]xTkQx2k + t + Sw(x1k) ] + ZL=-11Z2=1^,S (^k.SiD1(xk))+Z|=1*,g (pUD(xk.vk)) (57)

The gradient with respect to acceleration variables is:

VVkLD = 2G(Z)UT[G(Z)v + H(Z,Z)x2 + G(Z)]hk + zT^k

+ hk [S|=1Vvk*^g (pk.gjD(xk,vk,hk))] (58)

^5x5 hkI5x5 05x5 ^5x5

vk, k = 0, 2, ..., N-1

In the previous equations

,hkI5x5j

Vxk*,g, and VVk^garecalculated using the numerical differentiation formulas in equations 45-49.

5. AL-ANFIS Based Motion Planning

5.1 AL-ANFISSet up

Based on the result of the AL solution, an ANFIS-based structure is built to solve the online trajectory planning of the MPM. In [11, 23-24], an ANFIS-based inverse kinematic solver was proposed. In this paper we extend this principle to multi-objective planning, including plateform dynamic, torques and other technological constraints, in order to build an online ANFIS Planner. Both offline and online planners show that the trajectory planning of MPM is derived with small, acceptable error.A set of derivatives of (qi, q2, q3, qt, q5) is used to construct the true derivatives (x, y, z, q>, xn), and thus to obtain an error on which to apply the back-propagation algorithm. As mentioned above, here we add Xn related to qsfrom equation (15) to remove the redundancy of the system.

5.2 AL-ANFISNeuro-Fuzzy Inverse Planning

AL-ANFIS is a multi-layer feedforward adaptive network. The first layer is a two-input layer, characterizing the Cartesian position crisp values. The last layer is a three-outputlayer characterizing the corresponding crisp joint values (Figure 3). AL-ANFIS involves three hidden layers. The first is the fuzzification layer, which transfers the crisp inputs to linguistic variables through sigmoidal transfer functions. The second is the rule layer, which applies the product t-norm to produce the firing strengths of each rule. This is followed by a normalization layer. The training rule option is the Levenberg-Marquard version of the gradient back-propagation algorithm. This choice allows the learning process to be speeded up substantially with less iteration compared to standard back-propagation (e.g., gradient descent).

X y, Z, <t>, x„ Forward

Solution

fN \ TL, r„ I-J, T2, r3

Babtovard

Solution

Figure 3. AL-ANFIS Solution learning module 6. Simulation Results

The algorithm described in the previous section is built using Matlab software. The following simulation figures show different scenarios of minimizing time, energy, and both together. In time minimization, the energy weight is reduced to zero, and the same is done for energy minimization. For both time and energy minimizations the weight is set to 1. The following simulation figures show different scenarios of minimizing time, energy, and both together.

All the following simulation cases are performed for the initial values of theta are as follows:

ei = 0; er = 0; 0! = 1.3; 62 = 0.8; 63 = 1.4;

And the target values of thetas are:

eL = i; er = o.8; e^i.i; e2 = i.2; e3 = i.i.

6.1 Initial solution

To secure the convergence of the AL algorithm - even though it converges even if it starts from an unfeasible solution - a kinematically feasible solution is defined, based on an optimal time trajectory parameterization. The

initial time steps are assumed on an equidistant grid, for convenience:

hk = tk+1-tk=^,k=1,2.....N-1 (59)

Upon this parameterized minimum time trajectory, a predictive planning model is built in order to achieve a good initial solution for the AL.

To calculate the inertia matrix and the Coriolis and centrifugal dynamics components, we can use the approach developed initially for serial robots by Walker and Orin [25] based on the application of the Newton-Euler model of the robot dynamics. This method is straightforward and generallyapplicable to the case of MPM robots.

6.2 Search Direction Update

A limited-memory quasi-Newton method is used at each iteration of the optimization process to find the solution for the minimization step at the AL primal level, because the considered problem is of a large scale. In this research a systematic procedure similar to that used in [16] is used in the augmented Lagrangian implementation.

6.3 Simulated Scenario 1: Minimizing Time

Below are shown theseparate results of the minimization of time alone, energy alone, and both time and energy. Figures 4 to 7 show the simulated scenarios for minimizing time. Figures 8 to 11 show the simulated scenarios for minimizing energy. Figures 12 to 14 show the simulated scenarios for minimizing time and energy.

Figure 4 shows the variations of the end effector position due to minimization of time, and Figure 5 shows variations in the end effector position due to minimization of time. It also shows the variation of torque during the interval, and the variation of time steps along the path (Fig.6).

Figure 4. Variations of the end effector position due to minimization of time

' -0.005

t(sec)

t (sec)

t (sec)

-0.005

t (sec)

Figure 5. Variations of the end effector velocity due to minimization of time

02468 10 02468 10

t (sec)

t (sec)

0.15 0.1 N 0.05

-0.1-1-1-1-1- -0.051—

0 2 4 6 8 10 0 2 4 6 8 10

t (sec)

t (sec)

Figure 6. Variations of the torque due to minimization of time

6.4 Simulated Scenario 2: Minimizing Energy

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

Figure 8. Variations of the angles due to minimization of energy

-0.05 -0.1 -0.15 -0.2 -0.25

02468 10 02468 10

0 -0.01 ■ -0.02 . -0.03 -0.04 -0.05

0 2 4 6 8 10 t (sec)

2 4 6 8 10 t (sec)

Figure 9. Variations of the end effector position due to minimization of energy

0123456789 10 t(sec)

Figure 7. Variations of the time steps due to minimization of time

All the figures show that the minimization of both h and v gives a result close to the desired values, with acceptably small error. Moreover, the theta figures differences between the desired values and the values achieved, which are very close to the target points.

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

0.03 0.02 ) 0.01 - 0 -0.01 -0.02

-0.01 -0.02 -0.03 -0.04 -0.05

Figure 10. Variations of the end effector velocity due to minimization of energy

t (sec)

t (sec)

t (sec)

* -0.1 -0.2 -0.3

Is " 0

4 6 t (sec)

46 t (sec)

0.03 0.02 • 0.01 t0 -0.01

0.5 )0

" -1 -1.5

46 t (sec)

46 t (sec)

Figure 11. Variations of the torque due to minimization of energy

6.5 Simulated Scenario 3:Minimizing time and energy

Figure 13. Variations of the time step due to minimization of both time and energy

-0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 -0.04

AL-ANFIS is a structure built using the Fuzzy Logic Toolbox in the Matlab software, which is used to construct an online trajectory plan (Fig. 3). The result of the offline trajectory planning is used to run 400 different trajectories; each one contains 20 points along the trajectory. This gives 8000 samples, among which 750 are considered for training; testing and validation are achieved using 400 entry samples.

Figure 16 shows the training performance for AL-ANFIS, which is interesting as it achieves a very small root mean square error (RMSE), less than 0.1 in less than 10 epochs. It should be noted that the configuration used for the learning is one among infinite possible solutions for each input. Figure 17 shows the difference between the real and estimated values of the joint angles of the 8000samples. Better fine-tuning of the ANFIS parameters should improve the accuracy of the matching between ANFIS outcomes and the AL-provided results. This is the subject of on-going work.

1.3 1.25 a 1.2 1.15 1.1

_ 0.5 S 0

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

0 2 4 6 8 10 t (sec)

Figure 12. Variations of the angles due to minimization of both time and energy

Figure 14. Variations of the end effector velocity due to minimization of both time and energy

0 --0.1 --0.2 --0.3 -

4 6 8 t (sec)

468 t (sec)

0.03 0.02 0.01 0 -0.01

0.5 r-0

■ -0.5 -' -1 --1.5 L

468 t (sec)

468 t (sec)

Figure 15. Variations of the torque due to minimization of both time and energy

0.0706 0.0706 •5 0.0706 I 0.0706 0.0706 0.0706

4 6 Epochs

0.0192 0.0192 0.0192 0.0192 0.0192

0.0319 0.0319 ^ 0.0319 | 0.0319 0.0319 0.0319

46 Epochs

Figure 16. AL-ANFIS performance - root mean square error output with respect to learning epochs

t (sec)

t (sec)

46 t (sec)

46 t (sec)

46 t (sec)

■S 5

i0 ■tl

§ -10

0 2000 4000 6000 8000 -g 0 2000 4000 6000 8000 samples -n- samples

| 200,-

I 0 ti

i -200

0 2000 4000 6000 8000 samples

0 2000 4000 6000 8000 samples

^ 100 £ S j§ 0

0 2000 4000 6000 8000 samples

Figure 17. AL-ANFIS performance - difference between real and estimated values of the MPM values

7. Conclusion

This paper has provided a kinematic and dynamic analysis of mobile parallel robots (MPR). The inverse kinematic model is derived along with the joint limit avoidance through redundancy resolution. According to their complexity, the inverse kinematic model of mobile parallel robots is difficult to derive. The position kinematics solutions are derived and the Jacobian matrix that relates output velocities to the actuated joint rates is generated. Multi-objective motion planning of the MPM is then conducted. The objective criteria include time-energy, while satisfying hard constraint limits and technological limitations of the torques, join angles, and structural singularity avoidance of the hybrid structure.

By resorting to a neuro-fuzzy structure, the inverse kinematic is obtained using ANFIS with an initial minimum time trajectory parameterization. The simulation results validate the effectiveness of the derived models. A case study MPR composed of a three-wheeled nonholonomic mobile platform and a 3-RRPaR translational parallel robot was used for this purpose. Simulations in different scenarios focusing on time alone, energy alone and both together have the effectiveness of the proposal.

8. Acknowledgement

This work is supported by King Abdulaziz City of Science and Technology under Grant # KAP 0625- 11 and by King Fahd University of Petroleum & Minerals under Grant # FT10028.

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