Complexity Analysis for Calculating the Jacobian Matrix of 6DOF Reconfigurable Machines

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ProcediaCIRP 17 ( 2014 )218 – 223

Variety Management in Manufacturing. Proceedings of the 47th CIRP Conference on Manufacturing

Systems

Complexity analysis for calculating the Jacobian matrix of 6DOF

reconfigurable machines

Monika Z. Filiposkaa*, Ana M. Djuricb, Waguih ElMaraghya

a b

University of Windsor, IMSE Department, 401 Sunset Avenue, Windsor, ON, N9B 3P4, Canada Wayne State University, Engineering Technology Division, Fourth St. Detroit, MI 48202, USA

* Corresponding author. Tel.: +1-519-253-3000; +1-519-992-4565. E-mail address: filipos@uwindsor.ca

Abstract

The Jacobian matrix for machinery systems is challenging due to the kinematic structure, the machine behaviour, the machine configurations, and the singularity conditions. In the area of singularity, small velocities in the operational space may result in large velocities in the joint space. Therefore the control processes of machines may sometimes run into difficulties, since the inverse mapping from a Cartesian space to a joint space may cause problems. For solving the singularity of the kinematic structure, the development of the Jacobian matrix is the first priority. All of the different methods used for the Jacobian calculation are complex and require mathematical tools for symbolic calculation. It is important to select the most appropriate method with the minimum complexity for quick and accurate Jacobian matrix calculation. Thus, the authors were motivated to choose different procedures for developing the Jacobian matrix. In this study, the most general reconfigurable machinery kinematic structure, a remodelled n-DOF Global Kinematic Model, CNC-R GKM, is selected for the Jacobian matrix computation analysis. Each joint represents a combination of either rotational and/or translational joints type with any joint positive direction, with total number of 36 possible configurations in one joint. Calculation of the Jacobian matrix for this highly complex reconfigurable kinematic model, gives a comprehensive and unique results. The selected methods for Jacobian calculation are recursive Newton-Euler method and Vector cross multiplication method. The symbolic mathematical tool MAPLE 16 has been used. The comparison between these two calculation procedures has been completed with respect to the calculation complexity, and the results have been validated using the new developed CNC-Robot Global Kinematic Model (CNC-R GKM). The kinematic model and Jacobian matrix calculation used in this study are applicable in reconfigurable robotic and machines in industry, space and health care.

© 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2014 The Authors. Published by Elsevier B.V.

(http://wendang.chazidian.com/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the International Scientific Committee of “The 47th CIRP Conference on Manufacturing Selection and peer-review under responsibility of the International Scienti c Committee of “The 47th CIRP Conference on Systems” in the person of the Conference Chair Professor Hoda ElMaraghy.

Manufacturing Systems” in the person of the Conference Chair Professor Hoda ElMaraghy”

Keywords: Jacobian matrix; singularities; reconfigurable models;Newton-Euler; Vector method; complexity; robots.

1. Introduction

In today's enormous changing world, we struggle with the need to adapt to the changes quickly. Industries and manufacturers are dealing with the variety and diversity of demand. Customers are more and more insisting on unique products, which will satisfy their needs. For this to be accomplished, industries have to implement and employ reconfigurable systems, which will respond adequate and will be sustainable in today's inventive world [1].

Modelling a robotic system is a necessary premise to find appropriate control approaches. The kinematic of any mechanical system is primarily important. These systems, like industrial robots and CNC or rapid prototyping machines, usually have a problem with large displacements, and that is why they are experiencing large variations in geometric configuration when they operate even under normal conditions. The Jacobian matrix of any machine or

2212-8271 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://wendang.chazidian.com/licenses/by-nc-nd/3.0/).

Selection and peer-review under responsibility of the International Scientific Committee of “The 47th CIRP Conference on Manufacturing Systems” in the person of the Conference Chair Professor Hoda ElMaraghy” doi: 10.1016/j.procir.2014.02.051

Monika Z. Filiposka et al. / Procedia CIRP 17 ( 2014 ) 218 – 223 219

manipulator plays a key role in solving inverse kinematics

problem, but unfortunately, it loses its full rank at singularities. Although there are many algorithms for solving Jacobian matrix, still problems arise when a singular configuration is encountered on the way to a prescribed location in the task space. Newton algorithms are commonly used to solve Jacobian matrix for non-redundant robots. The only effective approach to handle the singularities is to keep a current configuration far away from singular configurations. In recent years operating speeds have been increased, and consequently, there has been an increase in accelerations and inertia forces. In theory, robot manipulators are flexible and can be reprogrammed for new tasks, but each robot's 0

R6-Rotational matrix

-Rotational transpose matrix i 1-Position matrix Pi

(0Ri)T

Zi 1-Unit vector

J - Jacobian matrix

-End effector velocity X

- Joint Velocity q

m- number of independent equations

n - number of links, number of joint variables

1.1. Literature survey configuration makes it capable for only a limited number of applications.

An n-DOF Global Kinematic Model (n-GKM) was previously developed [5] for any combination of either rotational or translational type of joints and it represents n-DOF kinematic structure with either rotational or translational joints. This unified approach, in principle, can be extended to manipulator’s architecture by including the design parameters. In this study a remodelled 6-DOF Global Kinematic Model (CNC-R GKM) is selected for the Jacobian matrix computation analysis. In this case, instead of calculating Jacobian for every robot and solving singularities separately, a reconfigurable Jacobian matrix has been developed. This calculation depends on the complexity of the multibody system, and iterative methods will have to be employed in the most difficult cases. The selected methods for Jacobian calculation are recursive Newton-Euler method and Vector method. The emphasis is on the significance of DH parameters for designing and building reconfigurable multibody system for industry. The solutions can be applied to any industrial robot, CNC or rapid prototyping machine, for calculating singular condition, consequently optimizing the same, and granting sustainable solution for the machinery control.

Nomenclature

CNC-R GKM - CNC- Robot global kinematic model D-H - Denavit-Hartenberg parameters DOF- Degrees of freedom

i 1

Ai- Homogeneous matrices

di- Link offset along previous Z to the common normal

Ti- Joint angle about Z, from old X to new X

ai- Link length of the common normal

Di- Twist angle about common normal, from old Z axis to new Z axis

Ri- Joints types control parameters Ti- Joints types control parameters

KSi- Twist angles sinus control parameter KCi- Twist angles cosines control parameter

i 1

T i- Angular velocity vector i 1P i- Linear velocity vector

i(0Zi)-Angular velocity i(0

Vi)-Linear velocity

The complexity in solving kinematics using direct computation is arising from the non-linearity of forward kinematics. Upon solving the kinematics for a system, the Jacobian matrix is the next priority, for determining singular configurations [2]. The Jacobian matrix calculation based methods are the most truthful, because they are giving the transformation between velocities in the workspace and the kinematic structure. Fast solutions can be obtained, due to the complexity for defining singularity configurations of reconfigurable structures. Different authors are offering unified approaches for Jacobian matrix calculation, but all of the methods differ in complexity and accuracy. In [5, 6] the authors developed the novel n-DOF Global Kinematic Model (GKM) for any combination of joints. The structure belongs to any of the three surfaces or eight sub surfaces. The total number of supported structures is48(48 1)n 1; for a 6DOF system, equals to 11,008,560,336 possible kinematic structures. The model is validated with 2DOF RR, RT, TR and TT structures. The same authors developed Reconfigurable Puma-Fanuc kinematic model [7]. The Jacobian matrices are calculated for all eight possible configurations of a 3 DOF kinematic structures and the comparison has been done according to the structures complexity in [4]. The proposed catalogue can be used as a design tool and for the validation of the reconfigurable based solutions. With complex configurations, the difficulty in calculation rises, thus beside the use of Maple Software, in most cases manual simplification is needed. Djuric et al [8] explained Newton-Euler Recursive method for Jacobian matrix calculation, also graphical approach for singularity analysis is included. Validation is conducted on Fanuc family robots, but this work can be extended for CNC-R GKM Jacobian matrix and singularity analysis, further to be incorporated in any robot configuration. In [9] several Newton based algorithms for inverse kinematics are presented and compared. All of them include a Jacobian derivation as compulsory. An approximation to the Jacobian matrix is not recommended by the authors, because of its weight in further recursive computation and trajectory tracking. Duleba and Sasiadek [10] present modified Jacobian method of transversal passing through singularity configurations for both redundant and non-redundant manipulators, but the Jacobian in this case is extended, although it still has square form. The method is not applicable for hyperbolic singularities, only for common quadratic singularities. It enables passing through singularities with smooth velocities, without stoppages or jerking. The method is evaluated on 3DOF manipulators.

220 Monika Z. Filiposka et al. / Procedia CIRP 17 ( 2014 ) 218 – 223

Abele et al [11] is describing a method for calculating

Cartesian stiffness based on Jacobian matrix. Even this method allows calculating realistic values for position and orientation in the entire Cartesian workspace, it is only for industrial robots with three degrees of freedom, and it can increase complexity when the workspace is bigger. Kircanski [12] describes simple Stanford manipulator inverse kinematic solution, where Jacobian is decomposed in four sub matrices [3x3], similar like in the Newton-Euler method, which reduces computation complexity 5-6 times. The problem of Jacobian calculation can be managed with designing the manipulators in order to have closed-form of solutions, but that leads to significant limitations in their applications. If no close-form solutions are available, approximations are Ri and Ti are used to control the selection of joint type

(rotational and/or translational). Their sinus and cosines are defined as the joint’s reconfigurable parameters and expressed in equations (4) and (5).

Ksi sinDi

(4)

Kci cosDi (5)

The need of being able to combine any robot manipulator and any CNC machine D-H parameters (Fig 1) has resulted in development of CNC-R Global Kinematic Model, graphically presented in Fig 2. In the literature, there is no evidence of kinematic structure that unifies different joint types (rotational and translational), and their Jacobian matrix solutions. conducted. Lenarcic and Kosutnik [13] is introducing concept for Jacobian and inverse kinematics approximation, applicable in arc welding robots, and validated on 6DOF. The method has proven accuracy, but in specific cases may introduce unsolvable problems in real-time computation. In [14] control equations based on manipulator Jacobian matrix are presented, which highlights its importance in modelling, kinematics, dynamics and control of robotic multibodies. Also [15] is raising the importance of Jacobian matrix in dynamics, thus unified generation of sparse Jacobian matrix for complex multibody systems is obtained. Another approach is developed with rearranging the Jacobian matrix [16, 17], based on screw theory. The manipulators (Puma type of robots) are always in singular configuration, but still can track a trajectory. Methods like in [18, 19] are offering reduced complexity in computation of the inverse Jacobian matrices. A classification of singularity configurations is offered in [20], where the chosen model for validation is Kuka KR 15/2. After a comparison between Jacobian method and null space approach for trajectory tracking, the results are showing that Jacobian method is simpler. In all reviewed papers the Denavit–Hartenberg (D-H) parameters [3] are used for the kinematic modelling.

2. Development of CNC-R GKM

In serial link manipulators there are series of links, connecting the end-effector to the base, by actuated joints. The homogeneous transformation matrix i 1Ai, in the n-DOF GKM [5] is giving the relationship between two joints in Cartesian coordinate frame, and i represents the number of joints (1, 2,...6).

ª«cos(RiTi TiTDHi) KCisin(RiTi TiTDHi)KSisin(RiTi TiTDHi)aicos(RiTi TiTDHi)ºi 1

Asin(RT TT)Kcos(RT Ti «

iiDHiCiiiiTDHi) KSicos(RiTi TiTDHi)aisin(RiTi TiTDHi)»

i«»«

0KSiKCiRidDHi Tidi»

¬

0001»¼i 1,2,3,4,5,6 (1)

Because each joint has six different positive directions for rotations or translations, any joint’s vector can be in positive or negative direction in Cartesian space. Thus, this model requires development of its own Jacobian matrix.

Rotational Joints: Ri 1,Ti 0 (2) Translational Joints: Ri 0,Ti 1 (3)

Combined joint types increase the model complexity, but provide the knowledge of many machines kinematic, which can be used as a design tool for new machine kinematic structure, and much more.

Table a) Robot manipulator D-H parameters

Mitsubishi PA10-6C D-H Parameters Joint Theta D A Alpha 1 0 317 0 -90 2 -90 0 450 0 3 90 0 0 90 4

0 480 0 -90

5 0 0 0 90 6

70

a)

+

Table b) CNC machine D-H parameters

CNC machine D-H Parameters

Joint Theta D A Alpha 1 90 4715.22 0 90 2 90 2714.59 0 90 3 0 0 0 0 4 180 -2664.68

0 -90

5 0 0 0 90 b)

6

-281.25

Fig. 1. Combination of CNC machine b) and robot manipulator a) kinematic

structures

The CNC-R GKM has possible kinematic configurations, and each configuration can be modelled with one set of reconfigurable parameters, presented in Table 1 and 2. Compared to the initial 6-DOF GKM, which can have 48(48 1)n 1

configurations, this kinematic model cannot

encompass all 11,008,560,336 structures, but includes 446,071,500 kinematic configurations.

Monika Z. Filiposka et al. / Procedia CIRP 17 ( 2014 ) 218 – 223

221

222 Monika Z. Filiposka et al. / Procedia CIRP 17 ( 2014 ) 218 – 223

added in the calculations, because of the complexity, also for

accuracy, depending on the type of joints (equations 10, 11).

i0

)] i 1,2, ,6 (10) (Zi) iRi 1[i 1(0Zi 1) Ri(i 1Ti

(Vi) iRi 1i 1(0Vi 1) Rii(0Zi)uiRi 1i 1Pi

i0

RiRi 1

^

i

>

>

i 1

i 1(0Z)ui 1PPii 1i

@`

@

i 1,2, ,6 (11)

Rotational matrices in these calculations, iRi 1 are defined by the upper left [3x3] sub matrices from the homogenous transformation matrices i 1Ai (equation 1), correlated with The Jacobian matrix derived using the Vector cross

multiplication method is relative to the base frame.

4. Comparison of the Jacobian matrices derived by the

two methods

The Jacobian matrices derived by the use of the both methods are dependant of the rotational and translational components, respectively. Because the CNC-R model is very complex 6DOF kinematic structure, the results are complex, also. Moreover manual decoupling is always necessary, especially in Jacobian matrix calculations in order to simplify each joint and equation (12) is presenting rotational transpose matrix.

ªcos(RiTi TiTDHi)

sin(RiTi TiTDHi)0ºi

R0

T

i) « 1 (Ri« KCisin(RiTi TiTDHi)KCicos(RiTi TiTDHi)KSi»(12) «R»¬KSisin(iTi TiTDHi) KSicos(RiTi TiTDHi)KCi¼»

The position vectorsi 1Pi, are defined by the upper right

[3x1] sub matrices from the homogenous transformation matrices i 1Ai(equation 1), correlated with each joint. See equation (13).

ªaicos(RiTi TiTDHi)º

i 1

Pi ««aisin(RiTi TiTDHi)» (13) «»

¬

RidDHi Tidi»¼

The Jacobian matrix calculated using the Newton-Euler method is relative to the end-effector frame. In order to get the Jacobian relative to the base frame, this matrix needs to be multiplied with the rotational matrix0R6.

3.2. Vector cross multiplication method The Vector method [22, 23] offers simpler results, especially if only kinematics computation is required. The method does not require velocities computation (does not involve differentiation too), and it is based on link transformation matrices found in forward (direct) kinematics (equations 14, 15).

­ªZ 1i 1Xi °«Pnº

»

J q °¬Zi 1¼ i 1,2, ,6;n 6 (14)

i

®°ªZi 1º

°¯«

¬0»¼

J ª«z0u0pz1u(0pn 0p1)z2u(0pn 0p2

) zn 1u(0pn 0pn 1)

º

¬zn

z1z

2

zn 1»¼

(15)

Although, Zi 1 unit (generating) vectors computation is

needed (equation 16), along the motion of the joints expressed in base frame coordinates, also pose matrixPn, computation is required (equation 17).

Zi RiZ0

(16)

Pn R1(R2(...(Rn 2(Rn 1pn pn 1)pn 1 ...) p2) p1

i 1,2, ,6;n 6 (17)

the results derived using both, Newton-Euler and Vector methods.

The derived results in equations (18) and (19) are presenting unified solution for 6DOF CNC-R GKM; also, they clearly show the dependence of the joints, from rotational and translational components separately.

Equation (18) presents the Jacobian matrix JNE calculated using the Newton-Euler method. Equation (19) presents the Jacobian matrix JV calculated using the Vector method.

ªJJ

JJ0«ºª«J1121J1222J12J14J150»«R1T

1º

«RT »

»JJ250»»«22« »NE

««J31JJ23JJ24«JJ32J33J34J350»R3T«41»«R 3»

» «JJ42J43J44450J»«4T4»56«¬J5161J5262J5363J5464

J65

0»«R»¼«5T 5»

«RT »¬66» (18)

¼

ªJJ

J00ª«11

º«T1d

º«JJ12J130000»«T 1»

«2d2» «21«J

31

J2232J2333000»»«T »«000000»«3d3»«»«T4d 4»«000000»««¬

00

00

0»¼»«T5d »

5»«¬T6d »

6»¼

ªJJJJJJ

«11ª«JJ12J13J14J15J

16º«Rº

«21«0J22J23J24J25J

26»»

»«R1»J«2»V«0J32J33J34J35J

36»«R3»««R» (19)

«0J4252

J43J44J45J

46»

«4»«¬J610

J5354J55«R5»63

J64

J

56»»65

66»¼«¬R»6¼»

ª«JJJJJ1115Jºª«J12J13JJJ

16»1 «21«0J22J23J1424J25«TºJ

26»

»«T«0J32J33J34J35«2»»J

36»«T3»««0J

42J43J44J4546»

«T»52«¬J61

J54J

55J

«0

53J«T4»»63

J

64

65

J

56»»66»¼«¬T5»6»¼

The 6DOF CNC-R GKM kinematic model joints are represented using generalized coordinates (equation 20).

q

i RiTior/and Tidi (20)

By observing these two equations it is clear that they are

different. These two methods are using two different reference frames in the calculation procedure, which generate their