Proceedings of the 2011 IEEE International Conference on Robotics and Biomimetics December 7-11, 2011, Phuket, Thailand
Optimal Design of a Spatial Four Cable Driven Parallel Manipulator Arian Bahrami and Mansour Nikkhah Bahrami
Abstract—This paper presents simulation and optimization of a spatial four-cable robot. For various points on the paths, tension in the cables and stiffness of the robot are determined before optimization. By selecting a function related to the stiffness matrix as the objective function of genetic algorithm, vertical positions of the connection points on the base platform as the optimization variables for each point on the paths can be determined in a way that the related function is optimized and a tension reduction in all the cables can be observed. Finally, the results of using different objective functions are presented for three different criteria: workspace volume, kinematic performance indices and actuating energy of the robot. The results indicate which function has the largest workspace volume with tension reduction property in all the four cables, which has the highest values for the kinematic performance indices and which has the lowest actuating energy. Index Terms— Cable-driven manipulator, Optimization, Simulation, Stiffness, Genetic Algorithm
I. INTRODUCTION Cable array robots, also known as cable-suspended robots or cable-driven manipulators, are those robots with cables and winches. The end-effector is connected to the base platform by a number of cables. Cable array robots have several advantages compared to traditional serial or parallel robot. They have high ratio of payload to robot weight and a much larger workspace, appropriate for high speed applications [1], lifting heavy loads [2], camera positioning in sports stadiums [3] and many others. Some disadvantages of cable robots are cable interference and inaccuracies at the endeffector as a result of cable stretch. Based on the number of cables (m) and the number of degrees of freedom (n), there are three categories for the cable robots [4], i.e. the incompletely restrained cable robots (mn+1). The completely restrained four cable robot with point-mass endeffector is the main focus of this paper. Some studies have been done on kinematics and statics of this type of cable robot [5]. Dynamic analysis has been investigated by a number of researchers [6]. There has been some work in design and implementation of this type of robot [3, 7]. There are researches dealing with the optimal design parameters of cable robots but most optimization work focused on the workspace optimization of planar [8] , and spatial cable array robots to obtain the optimal Arian Bahrami and Mansour Nikkhah Bahrami are with the Department of Mechanical Engineering, University of Tehran, Tehran, Iran (corresponding author to provide phone: +9821 8879 3257; fax: +9821 8807 9656; e-mail:
[email protected]).
978-1-4577-2137-3/11/$26.00 © 2011 IEEE
parameters of the robot [9]. To the best knowledge of authors, there are no proposals for a new type of spatial four cable robot with better performances in three different criteria (workspace volume, kinematic performance indices and actuating energy of the robot) compared with the conventional spatial four cable robots. Different optimization techniques have been reported in the literatures. Genetic algorithms (GAs) are one of the optimization techniques which work based on the principles of natural evolution. The GAs are appropriate for rapid global search of large and nonlinear spaces. Also, GAs are more useful for the discontinuous problem unlike the conventional gradientbased algorithms. However, GAs face with serious problems, typically premature convergence problems. Further description of GAs can be found in [10]. This paper proposes an optimal design of a spatial four cable robot.The organization of this paper is as follows: Section II describes the equations and variables used in robot modeling. In section III, by using Matlab software, robot motion before and after optimization for specified paths is simulated. In section IV, different kinds of objective functions are also introduced for the optimization problem. Section V deals with the results of the workspace, the kinematic performance indices and the actuating energy of each objective function to indicate which function has the best performance. II. MODELING The cable-array robot has a fixed square shaped platform, a moving point-mass end-effector and four cables each connected to a single connecting point on the end-effector and one of the vertexes of the square platform, allowing three degrees of freedom for the end-effector. OF is considered as the origin of the fixed coordinates XYZ which coincides with the mass center of the base platform (BP). A lumped mass located at the intersection point of all the cables is assumed for modelling the end-effector. Therefore, OM is the moving point-mass end-effector which coincides with coordinate xyz. rbase is the radius and the distance from OF to the connection points on the base platform Bi. The applied diagram of the model is shown in Fig. 1. This expression can be obtained from Fig.1:
li p bi
i 1, 2,3, 4
(1)
Where bi is the position vector of Bi, p is the position vector of point OM and li is the cable length vector from Bi to OM. The posture of the moving point mass end-effector (OM) is:
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some cases, this solution leads to negative values. Then the problem to solve is:
Z X
OF
bi
Y
Bi
p li
z x OM
y
Fig. 1. Vector polygon diagram of the robot
p X
Y
T
Z
(2)
min sT s (9) T under J s F mp s i & 0 1, 2,3, 4 ext i This problem is solved with the use of quadratic programming. For computing the minimum 2-norm solution, quadprog solver in Matlab’s Optimization Tool will be used in this paper [12]. The end-effector is deflected away from its desired position in the presence of the external forces. Increasing the robot stiffness leads to a reduction in deflection of the endeffector. The overall stiffness matrix Ku of the cable array robot can be defined as
The cable length qi can be defined as:
Ku J T diag (k1 , k2 , k3 , k4 ) J
qi li (liT li )1/ 2
(3)
Jacobian is a matrix which relates the end-effector velocity p to the cable velocity vector q [q1 q2 q3 q 4 ]T , written as: q Jp
(4)
The dynamic equation of the end-effector motion can be defined as: J T s Fext mp (5) Where
s [ s1
s
s2
is
s3
the
vector
of
cable
tensions
T
s 4 ] , Fext is the vector of external forces
applied to the end-effector Fext [ Fx
Fy
Fz ]T and m is
AE ki i i qi
J
i
l i qi
i 1, 2,3, 4
(6)
There exist an infinite number of solutions for Eq. (5). Therefore, the optimal tension distribution for this type of robot is important and has been presented by a number of researchers. There is some work based on linear programming (LP) in computing the minimum 1-norm solution [6]. Also, some researchers have suggested the use of quadratic programming (QP) to calculate the minimum 2norm solution [11]. Unlike QP methods, LP methods are prone to discontinuities in cable tensions for trajectory problems which may result in high frequency modes in the cables [7]. The minimum 2-norm solution can be obtained as: (7) ) s ( J T ) I ( Fext mp
(11)
For simulating the motions of the robot, these assumptions are considered as follows: 1 rbase 8.5m T
2 Fext 0 0 100 kN 3 E=200 Gpa
4 A=0.0004 m2 A. Simulation The specific paths are selected as shown in Table I. In simulation, a constant low speed for the robot is assumed, thus the acceleration of the end-effector is neglected. The robot motion before optimization for those paths is shown in Fig. 2. For those paths, the tension in all four cables and the Z stiffness values of the robot are calculated in the case of before optimization.
Where (JT)I is the generalized inverse of JT, defined as: ( J T ) I J ( J T J ) 1
i 1, 2,3, 4
In which Ei is the elasticity module of the cables, Ai is the area of the cross section of the cables and qi is the cable length. III. MATLAB RESULTS
the mass of the end-effector. The transpose of Jacobian matrix of the cable robot is written such that its ith column is T
(10)
Where
(8)
In a cable robot, all the components of s must be positive. In 2144
TABLE I THE SPECIFIC PATHS OF THE ROBOT Path No. 1 2 3 4 5 6 7 8 (reversed7) 9(reversed 6) 10(reversed 5) 11(reversed 4) 12(reversed 3) 13(reversed 2) 14(reversed 1)
X(m) 5.6 5.6 to 1.6 1.6 1.6 to 3.6 3.6 3.6 to -0.4 -0.4 -0.4 -0.4 to 3.6 3.6 3.6 to 1.6 1.6 1.6 to 5.6 5.6
Y(m) 1 to 0 0 0 0 0 to 2 2 2 2 2 2 to 0 0 0 0 0 to 1
Z(m) -8 -8 -8 to -4 -4 -4 -4 -4 to -3 -3 to -4 -4 -4 -4 -4 to -8 -8 -8
Time(s) 0-5 5-15 15-25 25-30 30-35 35-45 45-50 50-55 55-65 65-70 70-75 75-85 85-95 95-100
always minimizes the objective function. Therefore, for maximizing the Z stiffness, this function with a negative sign is used.
Robot motion for specific trajectory
0
C. Results of Simulation and Optimization
-2
Z(m)
-4
The results of the second case (one variable optimization) are presented in this section. The robot motion is simulated based on time frames. The model of the robot after optimization has been presented in Fig. 3. The changes in Zbase before and after optimization are shown in Fig. 4. Figure 5 shows the changes in the Z stiffness (kZ) before and after optimization. Figures 6-9 show the tension in the cables. After increasing the Z stiffness, reduction in cable tensions can be observed, which is in the interests of the designers.
-6 -8 -10 8
6
4
2
0 Y(m)
-2
-4
-6
-8
-8
-6
-4
-2
0
2
4
6
8
X(m)
Fig. 2. Motion of the robot for the specified paths (before optimization)
B. Optimization Using Genetic Algorithm
TABLE II GENETIC ALGORITHM PARAMETERS
Increasing the stiffness of the cable robot is one of the purposes of the paper. For the optimization process, Genetic algorithm is performed by using Matlab’s Genetic Algorithm Tool [13]. Firstly, the objective function and the independent variables are specified. The objective function used in optimization is the Z stiffness of robot which is the element (3, 3) of the stiffness matrix Ku. For each posture of the endeffector, the function is maximized to obtain the vector of the objective function variables Zbase which includes the vertical positions of the four vertexes of the square base platform. After defining Zbase for each point on the paths, the stiffness in the Z direction and tension in the cables are calculated. Before optimization, the radius of the base platform is constant as rbase=8.5. After optimization, this value for the radius is also applied with different heights of the base platform Zbase during the motion of robot. This can be done by using vertical rails on the base, which go in the direction of the gravity vector through the four vertexes of the square base platform. Therefore, the robot optimization can be written as follows:
Population type Population size Maximum generation Creation function Selection function Mutation function Crossover function Lower bounds on Zbase Upper bounds on Zbase
Double vector 300 100 uniform stochastic uniform Gaussian(scale = 1 , shrink =1) Scattered [-40 -40 -40 -40]T (m) [+40 +40 +40 +40]T (m)
Fig. 3. Motion of the robot for the specified paths (after optimization)
max k Z (Zbase ) (12) In which kZ is the stiffness in the Z direction of the stiffness matrix Ku and Zbase is the vector of the optimization variables which includes the Z positions of the four connection points on the base platform Zbase [ Z B1 Z B2 Z B3 Z B4 ]T . For
z position of base platform vs. time 10 before optimization after optimization
9
optimization, two cases are considered as follows:
8 7
1-Four independent variables 2-One independent variable
z (m)
6
In the first case, the four vertical slides move independently. In the second case, the four vertical slides move simultaneously. In other words, they move exactly the same. Several parameters affect the performance of GA such as encoding strategy, genetic operators, selection method, crossover rate, mutation rate, population size, maximum number of generation and fitness function. The parameters for the Matlab’s GA are set as Table II. The Matlab’s GA
5 4 3 2 1 0
0
10
20
30
40
50 time(s)
60
70
80
90
100
Fig. 4. Variations in Zbase before and after optimization based on travel time
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7
2
z stiffness vs. time
x 10
Force of cable 4 vs. time 100 before optimization after optimization
1.6
80
1.4
70
1.2 1 0.8
50 40 30
0.4
20
0.2
10
0
10
20
30
40
50 time(s)
60
70
80
90
0
100
Fig. 5. Variations in the Z stiffness of robot based on travel time
0
10
20
30
40
50 time(s)
60
70
80
90
100
Fig. 9. Variations in the tension in cable 4 based on travel time
Force of cable 1 vs. time
IV. OBJECTIVE FUNCTIONS
100 before optimization after optimization
90
Several objective functions are also considered for the optimization problem. The condition number of the Jacobian matrix, the condition number of the stiffness matrix and the minimum eigen value of the stiffness matrix are the three objective functions for the optimization problem.
80 70 cable force(kN)
60
0.6
0
before optimization after optimization
90
cable force(kN)
stiffness(N/m)
1.8
60 50 40
A. Dexterity index
30 20 10 0
0
10
20
30
40
50 time(s)
60
70
80
90
100
Fig. 6. Variations in the tension in cable 1 based on travel time Force of cable 2 vs. time 100
The condition number of the Jacobian matrix is known to be a measure of the control accuracy of the robot [14]. Therefore, it is a very important index. The minimization of this number keeps the robot away from the singularity configurations. The condition number (k) of the Jacobian matrix J can be written as
before optimization after optimization
90
k(J )
80
cable force(kN)
70 60 50 40 30 20 10 0
0
10
20
30
40
50 time(s)
60
70
80
90
100
Fig. 7. Variations in the tension in cable 2 based on travel time Force of cable 3 vs. time 100
B. Stiffness index
80
cable force(kN)
70
Maximizing the stiffness in all directions is more appropriate rather than a particular direction [15]. Therefore, the stiffness index is a suitable measure to use in the design process. It can be computed based on the condition number of the stiffness matrix. The condition number (k) of the stiffness matrix Ku can be expressed as
60 50 40 30 20 10 0
(13)
Where Kl and Ks are the maximum and the minimum singular value of the Jacobian matrix respectively. The square root of the eigen values of J T J are the singular values of the Jacobian matrix. The condition number takes its values between unity and infinity. The Jacobian matrix J is well-conditioned and far away from singularities, if the condition number of J reaches 1. On the other hand as the condition number approaches infinity the matrix is illconditioned. This quantity can be computed at each point within the workspace volume of the robot.
before optimization after optimization
90
Kl Ks
0
10
20
30
40
50 time(s)
60
70
80
90
k ( Ku )
100
Fig. 8. Variations in the tension in cable 3 based on travel time
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El Es
(14)
TABLE III WORKSPACE VOLUME FOR ONE VARIABLE OPTIMIZATION
Where El and Es are the maximum and the minimum eigen value of the stiffness matrix respectively. The condition number takes its values between unity and infinity. This quantity can also be computed at each point within the workspace volume of the robot. C. Minimum stiffness index The stiffness matrix Ku has positive eigen values 0 ≤ ke1≤ ke2≤ ke3. The stiffness in any direction is bounded between the minimum and the maximum eigen values in such a way that
ke1
Fext
p
ke3
Max(kZ) Min(condition(Ku)) Min (condition (J)) Max(min(eigen(Ku)))
V. THE RESULTS OF OBJECTIVE FUNCTIONS The results of using these functions will be presented in this section divided in to three important subsections. Firstly, the workspace volume for each objective function will be calculated. Secondly, global dexterity and stiffness indices for different objective functions are evaluated. Finally, the actuating energy for the robot will be computed for each objective function. A. Workspace Volume results Static equilibrium workspace is defined by the number of points where the point mass end-effector can be placed while all the cables must be in positive tension. Therefore, at each point within the search volume if all the elements of the vector of the cable tensions obtained from below Eq. (16) have nonnegative values, that point belongs to static equilibrium workspace volume. (16)
For obtaining the static equilibrium workspace volume before and after optimization, it is necessary to make these assumptions: 1-The search volume is a cube with these dimensions: -10m≤Z≤0, -8m≤Y≤8m, -8m≤X≤8m 2-The step size for the search volume is 0.4m for all X, Y and Z axes. 3- rbase= 8.5m 4-Fext= [0 0 -100] T kN 5-Using Zbase=0 and obtained Zbase from optimization process for each point within the search volume in the case of before and after optimization respectively.
Type 1 workspace
17538 10651 10316 9004
Type 1 %
Type 2 workspace
73 44 43 38
2224 9306 9534 10896
TABLE IV WORKSPACE VOLUME FOR FOUR VARIABLE OPTIMIZATION Objective function
Type 1 workspace
Type 1 %
Type 2 workspace
Max(kZ) Min(condition(Ku)) Max(min(eigen(Ku))) Min (condition (J))
11117 9826 9544 9106
46 41 39 38
42 2894 3490 1364
(15)
If a manipulator is close to a singularity, the minimum eigen value ke1 approaches zero. So, the minimum eigen value ke1 is a natural measure which indicates how far away the robot is from singularities. Therefore, maximization of this quantity is an appropriate way of increasing the robot stiffness and avoiding singularities.
s J T Fext under si 0 i 1, 2,3, 4
Objective function
A program was created with the use of Matlab software which tests the points in the search space to define if they belong to the static equilibrium workspace volume of the robot. By setting the five mentioned assumptions and using Eq. (16), one can obtain this workspace volume of the robot before and after optimization. Before optimization, the number of points obtained as the workspace volume is 24025. After optimization, the number of points obtained as the workspace volume is also 24025 for all the objective functions in case of one and four variable optimizations. In Table III, under the condition of using simultaneous motions for all the four slides, the results of using different objective functions made with the use of the stiffness matrix Ku for optimization are presented. In Table III and Table IV, type 1 workspace volume is the workspace with reduction in tension in all of the four cables and type 2 workspace volume is the workspace with increase in tension in all of the four cables compared with before optimization case. It can be seen from Table III that the first function kZ has the largest type 1 workspace volume (about 73 % of the workspace volume before optimization) which is in the interests of designers to have a reduction in tension in all the cables. In Table IV, under the condition of using four independent slides, the results of using different objective functions are presented. It is clear from Table IV that the first function kZ has the largest type 1 workspace volume (about 46 % of the workspace volume before optimization) which is less than the condition of using simultaneous motions for all the four slides. It is apparent that the same principal is also valid for other functions, so it is proper to apply simultaneous motions for all the four slides. B. Global dexterity and stiffness indices results The global dexterity index (GDI) is a measure of kinematic dexterity of the robot over the whole workspace [16]. The global dexterity index (GDI) shows how far away the robot is from singularities. High values for GDI guarantee good
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performance with respect to force and velocity transmission [9]. It is based on the distribution of the condition number of the Jacobian matrix over the entire workspace volume and is represented by 1 ( )dV GDI V k (17) dV
V
Where V is the volume of the workspace and k is the condition number of the Jacobian matrix. The reciprocal of the condition number ranges in value from zero to one which imposes a bound on GDI between zero and one. It is difficult to obtain the exact solution to the integral. Therefore, a discrete version of GDI can be defined as n
GDI
TABLE VII PARAMETERS OF THE INTEGRAL
In Table VI, the global dexterity and stiffness indices are presented when the four slides move independently. It is clear that using kZ leads to a much higher reduction in both indices compared with the results of Table V and therefore it is not a suitable function to use. Using other functions result in a much higher increase in both indices compared with the results of Table V, so it is proper to apply independent motions for all the four slides.
(18)
i
i 1
C. Energy results
n
In this section, the actuating energy is calculated before and after optimization. Before optimization, the total actuating energy is the sum of the actuating energy of the four electric motors. For calculating the actuating energy of each motor, this expression can be used before and after optimization: tf
Emotori
Max(kZ) Min(condition(Ku)) Max(min(eigen(Ku))) Min (condition (J))
0.516 0.736 0.712 0.741
GDI Relative change % -4 + 36 + 32 + 37
GSI
0.498 0.766 0.724 0.759
GSI Relative change % -8 + 41 + 34 + 40
TABLE VI GLOBAL INDICES FOR FOUR VARIABLE OPTIMIZATION Objective function
GDI
Max(kZ) Min(condition(Ku)) Max(min(eigen(Ku))) Min (condition (J))
0.385 0.843 0.810 0.906
GDI Relative change % - 29 + 56 + 50 + 67
GSI
0.348 0.970 0.911 0.873
GSI Relative change % - 35 + 79 + 68 + 61
s
i
qi dt i 1, 2,3, 4
(19)
0
After optimization, the total actuating energy is the sum of the actuating energy of the four motors and actuating energy of the four vertical sliders. The actuating energy of each vertical slider can be defined as: (20)
TABLE V GLOBAL INDICES FOR ONE VARIABLE OPTIMIZATION GDI
Unit s s kg m/s2
1
(k )
Where n is the number of points obtained as the workspace volume of the robot The global stiffness index (GSI) can also be defined for all over the workspace volume in the same way as the global dexterity index (GDI). High GSI value guarantees a high stiffness all over the workspace of the robot. Before optimization, the global dexterity index of the robot is 0.5410 and the global stiffness index is 0.5418. Table V presents the global dexterity and stiffness indices in the case of after optimization for different objective functions when the four slides move simultaneously. It is obvious that using kZ leads to a reduction in both indices. It decreases the global dexterity index by 4 %. It can also decrease the global stiffness index by 8 % which is not very significant. It is apparent that an increase in both indices can be observed when the other objective functions are used instead of kZ.
Objective function
Value 100 0.02 100 9.8
Variable tf dt mslider g
tf Eslideri Fi Zbase (i ) dt 0 T where : Fi mslider ( g Z base (i )) J (3, i ) si i 1,.., 4
Where Fi is the actuating force for the slider, mslider is the slider mass including the mass of the motor, g is the acceleration due to gravity, dt is the time step and tf is the total travel time of the robot motion. Table VII shows the values of the parameters used in computing the numerical integrals. Before optimization, the total actuating energy is 3640.2 kJ. Table VIII presents the actuating energy in the case of after optimization for different objective functions when the four slides move simultaneously. It is obvious that using kZ has the maximum ability to reduce the total actuating energy for the defined paths. It can decrease the actuating energy by about 19 %.
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TABLE VIII ACTUATING ENERGY FOR ONE VARIABLE OPTIMIZATION Objective function Max(kZ) Min(condition(Ku)) Min(condition(J)) Max(min(eigen(Ku)))
Motors (kJ) 1363.7 2293.0 2400.3 2734.6
Sliders (kJ) 1594.5 1296.6 1348.0 1670.7
Total (kJ) 2958.2 3589.6 3748.3 4405.3
REFERENCES
Relative change % - 19 -2 +3 + 21
TABLE IX ACTUATING ENERGY FOR FOUR VARIABLE OPTIMIZATION Objective function Max(kZ) Min(condition(Ku)) Max(min(eigen(Ku))) Min(condition(J))
Motors (kJ) 5837.7 133310.4 85201.7 159790.6
Sliders (kJ) 5433.8 133450.3 85212.2 159760.6
Total (kJ) 11271.5 266760.7 170413.9 319551.2
Relative change % + 209 + 7228 + 4581 + 8678
In Table IX, the actuating energy is presented for different objective functions when the four sliders move independently. It can be observed that using different objective functions lead to a steep increase in the actuating energy compared with the results of Table VIII due to much larger displacement of the slides and much higher tension in the cables during the robot motion, so it is logical to apply simultaneous motions for all the four sliders.
VI. CONCLUSION The cable robots have a lower stiffness in comparison to the classical parallel robots. Therefore, increasing the stiffness of the robot by the use of the optimization process is highly important. The optimization can be carried out by the changes in the vertical position of the base platform (Zbase) which can be obtained by using vertical rails in the base. The results indicated that there is a close relation between the type 1 workspace and the actuating energy. Higher value for type 1 workspace causes a higher reduction in the total actuating energy. Also, the results based on the workspace volume and actuating energy showed that it is logical to apply simultaneous motions for all the four vertical slides. In this case, increasing the stiffness in the Z direction, mostly leads to a reduction in tension in all the cables in 73% of the workspace volume with almost unchanged performance indices and a 19% energy reduction for the defined paths.The minimization of each of the condition number results in a reduction in tension in all the cables in an average of 43% of the workspace volume with an average of 39% increase in both indices and almost unchanged actuating energy for the defined paths. Maximizing the minimum stiffness leads to a reduction in tension in all the cables in 38% of the workspace volume with a 32% increase in both indices and a 21% energy increase for the defined paths. So, it is in the interests of the designers to design this type of robot using these functions.
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