Robot manipulators trends and development 2010 Part 12 pps

Di Gregorio, R., and Parenti-Castelli, V., 1998, "A Translational 3-dof Parallel Manipulator," Advances in Robot Kinematics: Analysis and Control Lenarcic, J., and Husty, M.. Di Gregorio

Eq (23) Each of the three polynomials T1, T2, and T3 can be written as follows:

4

where Q i and R i are the quotient and the remainder of a polynomial division on T i through

the divisor T4 with respect to a given variable x j At every point where all T i vanish along

with T4, all remainders R i must vanish too Therefore the equation set

is always equivalent to Eq (23), along with the condition T 4 0 If R1 , R2, and R3 are

remainders of polynomial divisions with respect to variables x2, x1, and x1, respectively, R1 ,

R2, and R3 are polynomials of degree four in the two variables x2 and x3 Therefore the

equation set:

can be solved with a method similar to that used for spherical wrists Variable x1 can be

hidden in the coefficients, and a partial homogenization with respect to x2 and x3 yields a set

of three homogeneous equations in three unknowns of degree four A resultant polynomial

in x1 can then be found through classical elimination methods

a) b) Fig 10 a) Positive critical points of manipulator T1;

b) All critical points of manipulator T1

Fig 11 The steepest ascent and descent paths are all singularity-free

Unfortunately, in this way the condition T 4 0 has not been directly imposed: Eq.(28) is not completely equivalent to Eq.(27), which introduces extraneous solutions The author has found no way to factor out such extraneous solutions from the resultant polynomial, however they can be easily detected, for they do not satisfy the condition T 4 0

Once all real solutions have been found by numerically solving the resultant polynomial,

and all extraneous solutions have been cancelled, all critical points of J are known The

classification of critical points, and the determination of steepest ascent paths is then analogous to the one proposed in Section 4.1 for spherical wrists

Manipulator T1 is now considered as a numerical example According to the conventional parameterization adopted before, T1, is defined by vectors reported in Table 2

(1,0,0) (0,1,0) (0,1,-1) (1,1,1) (0,1,-1) (1,1,1) (1,0,1)/ 2 (1,0,1)/ 2 (1,0,1)/ 2Table 2 Parameters defining manipulator T1

In the workspace of T1 there are three positive maxima and two positive 2-saddles, shown in Fig 10 a) Positive critical points of manipulator T1;Fig 10a through the conventional ball visualization proposed in Fig 9 Fig 10a also shows the steepest ascent paths (black lines), departing from the positive 2-saddles and reaching the maxima It can be seen that maxima

M2 and M3 are joined, while no paths reach maximum M1 Therefore there are two positive regions, free of parallel and constraint singularities

There are five negative minima and four negative 1-saddles, and the network of steepest descent paths is such that there are also two negative regions

Fig 10b shows all relevant critical points: the positive maxima are depicted as upward bound cones, the negative minima as downward bound cones, and the saddle points as spheres The network of singularity-free steepest ascent and descent paths is represented as black lines

Fig 11 shows two rotated views of the locus J 0 The outer spherical boundary belongs to the locus, but it has not been plotted, in order for the inside of the ball to be visible The darker surface inside the ball represents the locus of parallel singularities, whereas the brighter surface the locus of constraint singularities The intersection curve of the two surfaces is a set of singular degenerate critical points, that have been ruled out from the determination of critical points by means of the polynomial division It is possible to verify that the steepest ascent and descent paths never cross the spherical boundary, nor the parallel and constraint singularity loci

4.3 3RRR Planar manipulators

A 3RRR planar manipulator with general structure is depicted in Fig 12 The platform is connected to the rigid frame through three legs, composed of two connecting rods and three revolute joints, with the middle one actuated

The center of the ith leg revolute joint on the fixed frame is indicated by P i, whereas the

center of the ith leg revolute joint on the platform is indicated by Q i The center of the

actuated revolute joint of the ith leg is denoted by R i

Eq (23) Each of the three polynomials T1, T2, and T3 can be written as follows:

4

where Q i and R i are the quotient and the remainder of a polynomial division on T i through

the divisor T4 with respect to a given variable x j At every point where all T i vanish along

with T4, all remainders R i must vanish too Therefore the equation set

is always equivalent to Eq (23), along with the condition T 4 0 If R1 , R2, and R3 are

remainders of polynomial divisions with respect to variables x2, x1, and x1, respectively, R1 ,

R2, and R3 are polynomials of degree four in the two variables x2 and x3 Therefore the

equation set:

can be solved with a method similar to that used for spherical wrists Variable x1 can be

hidden in the coefficients, and a partial homogenization with respect to x2 and x3 yields a set

of three homogeneous equations in three unknowns of degree four A resultant polynomial

in x1 can then be found through classical elimination methods

a) b)

Fig 10 a) Positive critical points of manipulator T1;

b) All critical points of manipulator T1

Fig 11 The steepest ascent and descent paths are all singularity-free

Unfortunately, in this way the condition T 4 0 has not been directly imposed: Eq.(28) is not completely equivalent to Eq.(27), which introduces extraneous solutions The author has found no way to factor out such extraneous solutions from the resultant polynomial, however they can be easily detected, for they do not satisfy the condition T 4 0

Once all real solutions have been found by numerically solving the resultant polynomial,

and all extraneous solutions have been cancelled, all critical points of J are known The

classification of critical points, and the determination of steepest ascent paths is then analogous to the one proposed in Section 4.1 for spherical wrists

Manipulator T1 is now considered as a numerical example According to the conventional parameterization adopted before, T1, is defined by vectors reported in Table 2

(1,0,0) (0,1,0) (0,1,-1) (1,1,1) (0,1,-1) (1,1,1) (1,0,1)/ 2 (1,0,1)/ 2 (1,0,1)/ 2Table 2 Parameters defining manipulator T1

In the workspace of T1 there are three positive maxima and two positive 2-saddles, shown in Fig 10 a) Positive critical points of manipulator T1;Fig 10a through the conventional ball visualization proposed in Fig 9 Fig 10a also shows the steepest ascent paths (black lines), departing from the positive 2-saddles and reaching the maxima It can be seen that maxima

M2 and M3 are joined, while no paths reach maximum M1 Therefore there are two positive regions, free of parallel and constraint singularities

There are five negative minima and four negative 1-saddles, and the network of steepest descent paths is such that there are also two negative regions

Fig 10b shows all relevant critical points: the positive maxima are depicted as upward bound cones, the negative minima as downward bound cones, and the saddle points as spheres The network of singularity-free steepest ascent and descent paths is represented as black lines

Fig 11 shows two rotated views of the locus J 0 The outer spherical boundary belongs to the locus, but it has not been plotted, in order for the inside of the ball to be visible The darker surface inside the ball represents the locus of parallel singularities, whereas the brighter surface the locus of constraint singularities The intersection curve of the two surfaces is a set of singular degenerate critical points, that have been ruled out from the determination of critical points by means of the polynomial division It is possible to verify that the steepest ascent and descent paths never cross the spherical boundary, nor the parallel and constraint singularity loci

4.3 3RRR Planar manipulators

A 3RRR planar manipulator with general structure is depicted in Fig 12 The platform is connected to the rigid frame through three legs, composed of two connecting rods and three revolute joints, with the middle one actuated

The center of the ith leg revolute joint on the fixed frame is indicated by P i, whereas the

center of the ith leg revolute joint on the platform is indicated by Q i The center of the

actuated revolute joint of the ith leg is denoted by R i

The kinematic structure of the platform can be determined through the three parameters u2,

u3, and v3, defining the coordinates of Q1, Q2, and Q3 in the reference frame uQ1v attached to

the platform, as shown in Fig 12 Analogously, the kinematic structure of the fixed frame is

given by the three parameters a2, a3, and b3, defining the coordinates of P1, P2, and P3 in the

fixed reference frame xP1 y The ith leg can be defined through the lengths of the two

connecting rods: l i and m i (see Fig 12) Thus twelve parameters are used to define a 3RRR

manipulator

This class of planar manipulators have been widely studied, and often used as an example,

due to its simple kinematic architecture Workspace analysis methods for similar

manipulators were proposed in (Pennock & Kassner, 1993) and (Merlet et al.,1998) and the

singularity locus of analogous manipulators was defined and studied in (Sefrioui &

Gosselin, 1995) and (Wang & Gosselin, 1997)

The workspace of a 3RRR planar manipulator is a subset of the manifold containing all

possible positions of the platform in the plane Each point of the workspace will be

identified by the coordinates x and y of point Q1 in the fixed reference frame xP1y and by the

angle  between x- and u-axes

Fig 12 A 3RRR manipulator

The position of the ith actuator is given by the angle , between the two rods composing

each leg Any point in the jointspace is therefore identified by the three angles ( , , )  1 2 3

Any configuration of the manipulator can be represented through the six parameters

1 2 3

( , , , , , )x y    However, not any combination of these six parameters identifies a

configuration of the manipulator, for the ensuing constraints imposed by the three legs must

Eq.(29) can be easily derived by expressing the coordinates of each Q i in the fixed reference

frame xP1y, and by applying Carnot theorem to the three triangles P i Q i R i

The configuration space can be represented as the three dimensional manifold C described

by Eq (29) and embedded in the six dimensional manifold containing all the possible

vectors ( , , , , , )x y   1 2 3

Unlike the manipulators presented so far, the configuration space of planar 3RRR manipulators does not coincide with the workspace, and might be composed of more than one assembly configuration, therefore the proposed method will be applied to determine also the number of ACs and existence of feasible paths between any two configurations

In order to derive the equation of the singularity locus the relationship between the first order displacements of the platform and the actuators is needed Such relationship is obtained by differentiating Eq (29):

where s=(x, y,  ) and q=(1, 2, 3) Parallel singularities occur when the platform can

undergo infinitesimal displacements s, even though all actuators are locked, i.e q

vanishes Thus all singular points must satisfy the condition:

The singularity locus is a two-dimensional manifold defined by the zero level-set of the

function J , on the three-dimensional configuration space C

Lagrange’s optimization method is used again to find out critical points The Lagrangian

function L can be defined as:

Therefore, the following four cases are given

Case a): All Lagrange’s multipliers i are not equal to zero

In this case the sine of the three angles i must vanish (Eq.(34)), thus all three legs are completely outstretched or folded-up, for i is equal to 0 or  Such positions can be obtained

by substituting all possible combinations of 0 and  into each i of Eq.(29), which is

reobtained as derivatives of L with respect to Lagrange’s multipliers By subtracting the first equation of Eq.(29) from the last two, two linear equations in x and y are obtained From these linear equations, x and y can be determined as functions of the sine and cosine of , and back substituted into the first of Eq (29), yielding a trigonometric equation in , which

is easily solved through standard techniques Lagrange’s multipliers, which are useful for the classification of critical points, can be determined through the remaining derivatives of

L

Case b): ith Lagrange’s multiplier is equal to zero

In this case, only the sines of j and k vanish, with j and k different from i Analogous to the previous case, two equations for x, y, and  are obtained by substituting all possible

The kinematic structure of the platform can be determined through the three parameters u2,

u3, and v3, defining the coordinates of Q1, Q2, and Q3 in the reference frame uQ1v attached to

the platform, as shown in Fig 12 Analogously, the kinematic structure of the fixed frame is

given by the three parameters a2, a3, and b3, defining the coordinates of P1, P2, and P3 in the

fixed reference frame xP1 y The ith leg can be defined through the lengths of the two

connecting rods: l i and m i (see Fig 12) Thus twelve parameters are used to define a 3RRR

manipulator

This class of planar manipulators have been widely studied, and often used as an example,

due to its simple kinematic architecture Workspace analysis methods for similar

manipulators were proposed in (Pennock & Kassner, 1993) and (Merlet et al.,1998) and the

singularity locus of analogous manipulators was defined and studied in (Sefrioui &

Gosselin, 1995) and (Wang & Gosselin, 1997)

The workspace of a 3RRR planar manipulator is a subset of the manifold containing all

possible positions of the platform in the plane Each point of the workspace will be

identified by the coordinates x and y of point Q1 in the fixed reference frame xP1y and by the

angle  between x- and u-axes

Fig 12 A 3RRR manipulator

The position of the ith actuator is given by the angle , between the two rods composing

each leg Any point in the jointspace is therefore identified by the three angles ( , , )  1 2 3

Any configuration of the manipulator can be represented through the six parameters

1 2 3

( , , , , , )x y    However, not any combination of these six parameters identifies a

configuration of the manipulator, for the ensuing constraints imposed by the three legs must

Eq.(29) can be easily derived by expressing the coordinates of each Q i in the fixed reference

frame xP1y, and by applying Carnot theorem to the three triangles P i Q i R i

The configuration space can be represented as the three dimensional manifold C described

by Eq (29) and embedded in the six dimensional manifold containing all the possible

vectors ( , , , , , )x y   1 2 3

Unlike the manipulators presented so far, the configuration space of planar 3RRR manipulators does not coincide with the workspace, and might be composed of more than one assembly configuration, therefore the proposed method will be applied to determine also the number of ACs and existence of feasible paths between any two configurations

In order to derive the equation of the singularity locus the relationship between the first order displacements of the platform and the actuators is needed Such relationship is obtained by differentiating Eq (29):

where s=(x, y,  ) and q=(1, 2, 3) Parallel singularities occur when the platform can

undergo infinitesimal displacements s, even though all actuators are locked, i.e q

vanishes Thus all singular points must satisfy the condition:

The singularity locus is a two-dimensional manifold defined by the zero level-set of the

function J , on the three-dimensional configuration space C

Lagrange’s optimization method is used again to find out critical points The Lagrangian

function L can be defined as:

Therefore, the following four cases are given

Case a): All Lagrange’s multipliers i are not equal to zero

In this case the sine of the three angles i must vanish (Eq.(34)), thus all three legs are completely outstretched or folded-up, for i is equal to 0 or  Such positions can be obtained

by substituting all possible combinations of 0 and  into each i of Eq.(29), which is

reobtained as derivatives of L with respect to Lagrange’s multipliers By subtracting the first equation of Eq.(29) from the last two, two linear equations in x and y are obtained From these linear equations, x and y can be determined as functions of the sine and cosine of , and back substituted into the first of Eq (29), yielding a trigonometric equation in , which

is easily solved through standard techniques Lagrange’s multipliers, which are useful for the classification of critical points, can be determined through the remaining derivatives of

L

Case b): ith Lagrange’s multiplier is equal to zero

In this case, only the sines of j and k vanish, with j and k different from i Analogous to the previous case, two equations for x, y, and  are obtained by substituting all possible

combinations of 0 and  into the cosine of j and k , in the jth and the kth equations of Eq (29)

By subtracting one of such equations from the other, a linear equation in x is obtained:

( , ) ( , ) 0

which yields x as a function of y and  By equating to zero the derivatives of L with respect

to x, y, and , the ensuing equation is obtained:

( , , ) (1, , )T

j k

x y    

where A is a 33 matrix, whose columns contain the gradients of J , f j , and f k with respect to

variables x, y, and  Eq.(36) implies that the determinant of A must vanish, which yields the

third condition in x, y, and  By substituting the expression of x obtained from Eq.(35) into

this equation and into the jth equation of Eq.(29), the variable x is eliminated, and two

polynomial equations in y and the tangent of /2 are obtained, which can be easily solved

through Sylvester dyalitic elimination method (see Salmon, 1885) Among the solutions just

obtained, there are some extraneous solutions, which can be easily got rid of, for at such

solutions the two coefficients g and h of Eq.(35) vanish The angles j and k are equal to 0 or

, whilst the angle i can be derived from the ith equation of Eq (29) The jth and kth

Lagrange’s multipliers are obtained from Eq (36), and the ith is obviously zero

Case c): ith and jth Lagrange’s multipliers vanish

By equating to zero the derivatives of L with respect to x, y, and , the ensuing equation is

Where B is a 32 matrix, whose columns contain the gradients of J and fk with respect to

variables x, y, and  Eq.(37) implies that all the three 22 minors of B are singular, which

yields three equations By considering two of the three conditions just derived, along with

the equation obtained by substituting 0 or  into k in the kth equation of Eq (29), three

equations in the variables x, y, and  are obtained, which can be solved analogously to case

b) It is possible to prove that, by equating to zero only two of the three 22 minor

determinants, some extraneous solutions are introduced, which do not make the third

determinant vanish By imposing this last condition, it is possible get rid of such extraneous

solutions

Case d): All Lagrange’s multipliers are equal to zero

In this case the gradient of J with respect to x, y, and  must vanish, which yields two

linear equations in x and y, and a quadratic equation in x and y This equation set can be

solved by techniques analogous to case a)

Once the critical points are determined, they are all classified by means of the bordered

Hessian, as discussed in Section 4.1, and the maximum increase and decrease directions in

the neighbourhoods of the saddle points are determined

Two numerical examples are presented hereafter, manipulators P1 and P2 The kinematical

structure of the two examples is summarized in Table 3, according to the parameterization

adopted

Table 3 Parameters defining manipulators P1 and P2

In manipulator P1 there are four positive maxima, nine positive saddles, four negative saddles, and no negative maxima The four positive maxima are shown in Fig 13a Maxima

2-M1 and M2 are joined by steepest ascent paths starting from some of the positive 2-saddles,

while M3 and M4 are not connected to any other maximum by any steepest ascent path

starting from any positive or negative 2-saddle There are three ACs: one containing M1 and

M2, and the other two containing M3 and M4 Manipulator P1 was generated by imposing that the loop composed by leg 1, leg 2, the platform and the frame have two ACs, through the condition derived in (Foster & Cipra, 1998), and that leg 3 be able to completely outstretch in one of such ACs, but not in the other Therefore one of the two ACs of the loop

is split into two ACs by the fact that leg 3 can never outstretch, nor fold back

a) b) Fig 13 a) The four maxima of manipulator P1

b) Three maxima of manipulator P2 The analysis of the negative critical points shows that each of the tree ACs is split into two PSFRs, one positive, and one negative Therefore, if the sign of the Jacobian determinant is the same at two configurations belonging to the same AC, a singularity-free path connecting them always exists

However, the ACs are not always split into two PSFRs only, as manipulator P2 shows In manipulator P2 there is only one AC, therefore any configuration of the manipulator is reachable, but this AC is split into four PSFRs, three positive and one negative Fig 12b shows three positive maxima belonging to the three positive PSFRs: many feasible paths connect these three configurations where the Jacobian determinant is positive, but none of them is free of parallel singularities

combinations of 0 and  into the cosine of j and k , in the jth and the kth equations of Eq (29)

By subtracting one of such equations from the other, a linear equation in x is obtained:

( , ) ( , ) 0

which yields x as a function of y and  By equating to zero the derivatives of L with respect

to x, y, and , the ensuing equation is obtained:

( , , ) (1, , )T

j k

x y    

where A is a 33 matrix, whose columns contain the gradients of J , f j , and f k with respect to

variables x, y, and  Eq.(36) implies that the determinant of A must vanish, which yields the

third condition in x, y, and  By substituting the expression of x obtained from Eq.(35) into

this equation and into the jth equation of Eq.(29), the variable x is eliminated, and two

polynomial equations in y and the tangent of /2 are obtained, which can be easily solved

through Sylvester dyalitic elimination method (see Salmon, 1885) Among the solutions just

obtained, there are some extraneous solutions, which can be easily got rid of, for at such

solutions the two coefficients g and h of Eq.(35) vanish The angles j and k are equal to 0 or

, whilst the angle i can be derived from the ith equation of Eq (29) The jth and kth

Lagrange’s multipliers are obtained from Eq (36), and the ith is obviously zero

Case c): ith and jth Lagrange’s multipliers vanish

By equating to zero the derivatives of L with respect to x, y, and , the ensuing equation is

Where B is a 32 matrix, whose columns contain the gradients of J and fk with respect to

variables x, y, and  Eq.(37) implies that all the three 22 minors of B are singular, which

yields three equations By considering two of the three conditions just derived, along with

the equation obtained by substituting 0 or  into k in the kth equation of Eq (29), three

equations in the variables x, y, and  are obtained, which can be solved analogously to case

b) It is possible to prove that, by equating to zero only two of the three 22 minor

determinants, some extraneous solutions are introduced, which do not make the third

determinant vanish By imposing this last condition, it is possible get rid of such extraneous

solutions

Case d): All Lagrange’s multipliers are equal to zero

In this case the gradient of J with respect to x, y, and  must vanish, which yields two

linear equations in x and y, and a quadratic equation in x and y This equation set can be

solved by techniques analogous to case a)

Once the critical points are determined, they are all classified by means of the bordered

Hessian, as discussed in Section 4.1, and the maximum increase and decrease directions in

the neighbourhoods of the saddle points are determined

Two numerical examples are presented hereafter, manipulators P1 and P2 The kinematical

structure of the two examples is summarized in Table 3, according to the parameterization

adopted

Table 3 Parameters defining manipulators P1 and P2

In manipulator P1 there are four positive maxima, nine positive saddles, four negative saddles, and no negative maxima The four positive maxima are shown in Fig 13a Maxima

2-M1 and M2 are joined by steepest ascent paths starting from some of the positive 2-saddles,

while M3 and M4 are not connected to any other maximum by any steepest ascent path

starting from any positive or negative 2-saddle There are three ACs: one containing M1 and

M2, and the other two containing M3 and M4 Manipulator P1 was generated by imposing that the loop composed by leg 1, leg 2, the platform and the frame have two ACs, through the condition derived in (Foster & Cipra, 1998), and that leg 3 be able to completely outstretch in one of such ACs, but not in the other Therefore one of the two ACs of the loop

is split into two ACs by the fact that leg 3 can never outstretch, nor fold back

a) b) Fig 13 a) The four maxima of manipulator P1

b) Three maxima of manipulator P2 The analysis of the negative critical points shows that each of the tree ACs is split into two PSFRs, one positive, and one negative Therefore, if the sign of the Jacobian determinant is the same at two configurations belonging to the same AC, a singularity-free path connecting them always exists

However, the ACs are not always split into two PSFRs only, as manipulator P2 shows In manipulator P2 there is only one AC, therefore any configuration of the manipulator is reachable, but this AC is split into four PSFRs, three positive and one negative Fig 12b shows three positive maxima belonging to the three positive PSFRs: many feasible paths connect these three configurations where the Jacobian determinant is positive, but none of them is free of parallel singularities

7 Conclusion

This work presented a numerical method able to count and identify the PSFRs and the ACs

carved by the singularity locus in the configuration space of a manipulator, and its

application to three types of parallel manipulators

In principle, this method works for any manipulator, but some very particular cases, where

there are degenerate critical points of the Jacobian determinant The application is rather

simple, except the determination of all critical points of the Jacobian determinant on the

configuration space This part of the procedure reduces in most cases to the determination of

all solutions to a polynomial equation set, that might be a very hard task in practice,

although it is always theoretically possible

However, if the determination of the critical points of the Jacobian determinant is viable, like

the presented examples, the proposed method represents a stable and powerful tool for

analyzing the topology of the singularity locus and for planning singularity-free paths

The proposed method does not take into account the possible reduction of configuration

space of a manipulator due to the mechanical interference between the links, or by actuator

limits The analysis of the singularity locus under the additional constraint that no collision

between the links takes place is a possible future development of the proposed method, as

well as its application to more parallel manipulators with six degrees of freedom

8 References

Bhattacharya, S., Hatwal, H., and Ghosh, A., 1998, “Comparison of an exact and an

approximate method of singularity avoidance in platform type parallel

manipulators,” Mechanism and Machines Theory, 33(7), pp 965–974

Chablat, D and Wenger, P., 1998, “Working modes and aspects in fully parallel Leuven,

Belgium, pp 1964–1969

Chase, T.R and Mirth, J.A., 1993, “Circuits and branches of single-degree-of-freedom planar

linkages,” Journal of Mechanical Design, 115(2), pp 223-230

Dasgupta, B and Mruthyunjaya, T., 1998, “Singularity-free path planning for the Stewart

platform manipulator,” Mechanism and Machine Theory, 33(6), pp 711–725

Di Gregorio, R., and Parenti-Castelli, V., 1998, "A Translational 3-dof Parallel Manipulator,"

Advances in Robot Kinematics: Analysis and Control Lenarcic, J., and Husty, M L.,

Eds., Kluwer Academic Publishers, pp 49-58

Di Gregorio, R., and Parenti Castelli V., 2002, "Mobility Analysis of the 3-UPU Parallel

Mechanism Assembled for a Pure Translational Motion," Journal of mechanical

Design,Vol 124, pp 259-264

Dou, X and Ting, K.L., 1998, “Identification of singularity free joint rotation space of

two-dof parallel manipulators,” proceedings of DETC’98, Atlanta, Georgia

Fletcher, R.,1987, Practical Methods of Optimization, John Wiley & Sons, Chichester

Foster, D.E and Cipra, R.J., 1998, “Assembly configurations and branches of single-loop

mechanism with pin joints and sliding joints,” Journal of Mechanical Design,

120(3), pp 387-391

Foster, D.E and Cipra, R.J., 2002, “An automatic method for finding the assembly

configurations of planar non-single-input-dyadic mechanisms,” Journal of

Mechanical Design, 124(1), pp 58-67

Gosselin, C and Angeles, J., 1990, “Singularity analysis of closed-loop kinematic chains,”

IEEE Transactions On Robotics and Automation, 6(3), pp 281–290

Hirsch, M.W., 1976, Differential Topology, Springer, New York

Innocenti, C., and Parenti-Castelli, V , 1993, “Echelon form solution of direct Kinematics for

the general fully-parallel spherical wrist,” Mechanism and Machine Theory, vol 28, No.4, pp 553–561

Kevin Jui C.K and Qiao Sun, 2005, “Path tracking of parallel manipulators in the presence of

force singularity,” Journal of Dynamic Systems, Measurement and Control, 127, pp 550–563

Merlet, J.P., Gosselin, C.M., and Mouly, N., 1998, “Workspaces of planar parallel

manipulators,” Mechanism and Machine Theory, 33(1/2), pp.7-20

Milnor, J., 1969, Morse Theory, Princeton University Press

Mirth, J.A and Chase, T.R., 1993, “Circuit analysis of Watt chain six-bar mechanisms,”

Journal of Mechanical Design, 115(2), pp.214-222

Midha, A., Zhao, Z.L., and Her, I., 1985, “Mobility conditions for planar linkages using

triangle inequality and graphical interpretation,” Journal of Mechanical Design, 107(3), pp 394-400

Paganelli, D., 2008, “Topological Analysis of Singularity Loci for Serial and Parallel

Manipulators,”Phd Thesis, University of Bologna

Paul, B., 1979, “A reassessment of Grashof’s Criterion,” Journal of Mechanical Design, 101,

pp 515-518

Parenti-Castelli, V., Di Gregorio, R., and Lenarcic, J., 1998, "Sensitivity to Geometric

Parameter Variation of a 3-dof Fully-Parallel Manipulator," Proceedings of the 3rd International Conference on Advanced Mechatronics, pp 364-369

Pennock, G.R and Kassner, D.J., 1993, “The workspace of a general geometry planar

three-degree-of-freedom platform-type manipulator,” Journal of Mechanical Design, 115(1), pp 269-276

Salmon, G , 1885, Modern Higher Algebra, Hodges, Figgis & Co., Dublin

Sen, S , Dasgupta, B., and Mallik, A.K., 2003, “Variational approach for singularity-free

path-planning of parallel manipulators,” Mechanism and Machine Theory, 38, pp 1165–1183

Sefrioui, J., Gosselin, C., 1994, "Etude et Representation des Lieux de Singularite des

Manipulateurs Paralleles Spheriques a Trois Degres de Liberte avec Actionneurs Prismatiques," Mechanisms and Machines Theory, 29, No.4, pp.559-579

Sefrioui, J and Gosselin, C.M., 1995, “On the quadratic nature of the singularity curves of

planar three-degree-of freedom parallel manipulators,” Mechanism and Machine Theory, 30(4), pp.533-551

Tsai, L W., 1996, "Kinematics of a Three-dof Platform With Three Extensible Limbs," Recent

Advances in Robot Kinematics, Lenarcic, J., and Parenti-Castelli, V., Eds., Kluwer Academic Publishers,pp 401-410

Wang, J., and Gosselin, C.M., 1997, “Singularity loci of planar parallel manipulators with

revolute actuators,” Robotics and Autonomous Sustems, 21, pp.377-398

Whitehead, W., 1978, Elements of Homotopy Theory, Springer, New York

Zlatanov, D., Bonev I.A., and Gosselin, C.N., 2002, "Constraint Singularities of Parallel

Mechanisms," Proceedings of the 2002 IEEE International Conference on Robotics & Automation, pp 496-502

7 Conclusion

This work presented a numerical method able to count and identify the PSFRs and the ACs

carved by the singularity locus in the configuration space of a manipulator, and its

application to three types of parallel manipulators

In principle, this method works for any manipulator, but some very particular cases, where

there are degenerate critical points of the Jacobian determinant The application is rather

simple, except the determination of all critical points of the Jacobian determinant on the

configuration space This part of the procedure reduces in most cases to the determination of

all solutions to a polynomial equation set, that might be a very hard task in practice,

although it is always theoretically possible

However, if the determination of the critical points of the Jacobian determinant is viable, like

the presented examples, the proposed method represents a stable and powerful tool for

analyzing the topology of the singularity locus and for planning singularity-free paths

The proposed method does not take into account the possible reduction of configuration

space of a manipulator due to the mechanical interference between the links, or by actuator

limits The analysis of the singularity locus under the additional constraint that no collision

between the links takes place is a possible future development of the proposed method, as

well as its application to more parallel manipulators with six degrees of freedom

8 References

Bhattacharya, S., Hatwal, H., and Ghosh, A., 1998, “Comparison of an exact and an

approximate method of singularity avoidance in platform type parallel

manipulators,” Mechanism and Machines Theory, 33(7), pp 965–974

Chablat, D and Wenger, P., 1998, “Working modes and aspects in fully parallel Leuven,

Belgium, pp 1964–1969

Chase, T.R and Mirth, J.A., 1993, “Circuits and branches of single-degree-of-freedom planar

linkages,” Journal of Mechanical Design, 115(2), pp 223-230

Dasgupta, B and Mruthyunjaya, T., 1998, “Singularity-free path planning for the Stewart

platform manipulator,” Mechanism and Machine Theory, 33(6), pp 711–725

Di Gregorio, R., and Parenti-Castelli, V., 1998, "A Translational 3-dof Parallel Manipulator,"

Advances in Robot Kinematics: Analysis and Control Lenarcic, J., and Husty, M L.,

Eds., Kluwer Academic Publishers, pp 49-58

Di Gregorio, R., and Parenti Castelli V., 2002, "Mobility Analysis of the 3-UPU Parallel

Mechanism Assembled for a Pure Translational Motion," Journal of mechanical

Design,Vol 124, pp 259-264

Dou, X and Ting, K.L., 1998, “Identification of singularity free joint rotation space of

two-dof parallel manipulators,” proceedings of DETC’98, Atlanta, Georgia

Fletcher, R.,1987, Practical Methods of Optimization, John Wiley & Sons, Chichester

Foster, D.E and Cipra, R.J., 1998, “Assembly configurations and branches of single-loop

mechanism with pin joints and sliding joints,” Journal of Mechanical Design,

120(3), pp 387-391

Foster, D.E and Cipra, R.J., 2002, “An automatic method for finding the assembly

configurations of planar non-single-input-dyadic mechanisms,” Journal of

Mechanical Design, 124(1), pp 58-67

Gosselin, C and Angeles, J., 1990, “Singularity analysis of closed-loop kinematic chains,”

IEEE Transactions On Robotics and Automation, 6(3), pp 281–290

Hirsch, M.W., 1976, Differential Topology, Springer, New York

Innocenti, C., and Parenti-Castelli, V , 1993, “Echelon form solution of direct Kinematics for

the general fully-parallel spherical wrist,” Mechanism and Machine Theory, vol 28, No.4, pp 553–561

Kevin Jui C.K and Qiao Sun, 2005, “Path tracking of parallel manipulators in the presence of

force singularity,” Journal of Dynamic Systems, Measurement and Control, 127, pp 550–563

Merlet, J.P., Gosselin, C.M., and Mouly, N., 1998, “Workspaces of planar parallel

manipulators,” Mechanism and Machine Theory, 33(1/2), pp.7-20

Milnor, J., 1969, Morse Theory, Princeton University Press

Mirth, J.A and Chase, T.R., 1993, “Circuit analysis of Watt chain six-bar mechanisms,”

Journal of Mechanical Design, 115(2), pp.214-222

Midha, A., Zhao, Z.L., and Her, I., 1985, “Mobility conditions for planar linkages using

triangle inequality and graphical interpretation,” Journal of Mechanical Design, 107(3), pp 394-400

Paganelli, D., 2008, “Topological Analysis of Singularity Loci for Serial and Parallel

Manipulators,”Phd Thesis, University of Bologna

Paul, B., 1979, “A reassessment of Grashof’s Criterion,” Journal of Mechanical Design, 101,

pp 515-518

Parenti-Castelli, V., Di Gregorio, R., and Lenarcic, J., 1998, "Sensitivity to Geometric

Parameter Variation of a 3-dof Fully-Parallel Manipulator," Proceedings of the 3rd International Conference on Advanced Mechatronics, pp 364-369

Pennock, G.R and Kassner, D.J., 1993, “The workspace of a general geometry planar

three-degree-of-freedom platform-type manipulator,” Journal of Mechanical Design, 115(1), pp 269-276

Salmon, G , 1885, Modern Higher Algebra, Hodges, Figgis & Co., Dublin

Sen, S , Dasgupta, B., and Mallik, A.K., 2003, “Variational approach for singularity-free

path-planning of parallel manipulators,” Mechanism and Machine Theory, 38, pp 1165–1183

Sefrioui, J., Gosselin, C., 1994, "Etude et Representation des Lieux de Singularite des

Manipulateurs Paralleles Spheriques a Trois Degres de Liberte avec Actionneurs Prismatiques," Mechanisms and Machines Theory, 29, No.4, pp.559-579

Sefrioui, J and Gosselin, C.M., 1995, “On the quadratic nature of the singularity curves of

planar three-degree-of freedom parallel manipulators,” Mechanism and Machine Theory, 30(4), pp.533-551

Tsai, L W., 1996, "Kinematics of a Three-dof Platform With Three Extensible Limbs," Recent

Advances in Robot Kinematics, Lenarcic, J., and Parenti-Castelli, V., Eds., Kluwer Academic Publishers,pp 401-410

Wang, J., and Gosselin, C.M., 1997, “Singularity loci of planar parallel manipulators with

revolute actuators,” Robotics and Autonomous Sustems, 21, pp.377-398

Whitehead, W., 1978, Elements of Homotopy Theory, Springer, New York

Zlatanov, D., Bonev I.A., and Gosselin, C.N., 2002, "Constraint Singularities of Parallel

Mechanisms," Proceedings of the 2002 IEEE International Conference on Robotics & Automation, pp 496-502

Luis Hernández, Hichem Sahli and René González

0

Vision-based 2D and 3D Control of Robot Manipulators

1Universidad Central de Las Villas (UCLV),2Vrije Universiteit Brussel (VUB),

3Empresa de Automatización Integral (CEDAI)

1Cuba,2Belgium,3Cuba

1 Introduction

Robotics has been a paradigm for science in the last few decades At first, scientists’ efforts

were devoted to the solution of the problem of planning and control of the motion of robot

ma-nipulators However, the motion control of robot manipulators in unstructured environments

is today an attractive scientific problem An interesting solution for motion control is the

use of sensor information, such as computer vision, in the system’s feedback Several works

and tools have been developed in recent years in this field (Corke; 2005), (Chaumette and

Hutchinson; 2006) The more typical approaches consider visual perception for servoing and

for the so called look and move (Hutchinson et al.; 1996) Visual servoing (Kelly et al.; 2000) can

be classified into two approaches: camera-in-hand or camera-to-hand (Flandin et al.; 2000)

In camera-to-hand robotic systems, multiple cameras or a single camera fixed in the

world-coordinate frame capture images of both, the robot and its environment The tracking of the

object with visual feedback can be made in 2D or 3D An interesting solution for the visual

servoing of camera-in hand robot manipulators in 2D can be found in e g (Bonfe et al.; 2002)

and (Hernández et al.; 2008) where stability demonstration of a decoupled controller have

been presented For 3D tracking some solutions reported are look and move controller, with

one camera (Sim et al.; 2002) or more than one camera (Xie et al.; 2005) A 3D visual servoing

with stability analysis in continuous time is presented by (Hernández et al.; 2008a) and (Kelly

et al.; 2006) present a direct visual servoing with transpose Jacobian control technique for

reg-ulation of robot manipulators in the 3D Cartesian space In a similar vein (Enescu et al.; 2006)

present mobile robot navigation for person tracking using a stereo head-camera

In this chapter, we consider the control problem of camera-in-hand robot manipulators in 2D

and 3D In both cases only one camera is mounted on the robot’s arm, which supplies visual

information of the environment, with the aim of moving the manipulator by maintaining the

image of the tracked object (a sphere) in the centre of the image plane, despite the possible

movements of the object In the 3D control the constant radius is used as a feature too In this

work, the proposed control system considers two loops in cascade, an internal loop solving

the robots’ joint control, and an external loop implementing a dynamic look and move visual

controller A stability analysis in discrete time is developed under the conditions that it is

possible to approximate the dynamic effect of the internal loop as an external loop time delay

(Corke; 1996) and (Bonfe et al.; 2002) The more classic presentations of servovisual control

are velocity controllers, based in the term feature Jacobian,(Chaumette and Hutchinson; 2006)

20

Fig 1 Camera-in-hand robotic system

(Chaumette and Hutchinson; 2007), but we present the robotic and vision systems modeled

for small variations about the operating point for position control and in these conditions the

stability of the whole system are balanced

A particular study is made using an ASEA IRB6 robot manipulator which has mechanically

decoupled wrist This allows keeping the orientation of the camera’s optical axis while the

arm is moving

In order to validate the proposed control system, a general stability analysis is presented and

an analysis of the step response in a regulator type system is made The disturbance is

inter-preted as initial conditions To illustrate the proposed controller, the control system stability

and its performance, both simulation results as well as experimental results using the ASEA

IRB6 robot manipulator are presented Our experimental results confirm the expected step

response in the image plane, with good time performance and zero steady-state error

2 Robotic System Model

As shown in Fig 1, the robotic system considered has a robot manipulator with a camera in its

hand The basic mathematical description of this system consists of the robot and the camera

model

2.1 Robot Kinematics Model

The kinematics of a manipulator gives the relationship between the joint positions q and the

corresponding tool translational(x, y, z) and angular position(α, β, γ) For an n-axis rigid

link manipulator, the forward kinematic solution, T, could be computed for any manipulator,

irrespective of the number of joints or kinematic structure (Barrientos et al.; 1997) A genericmathematical representation could be:

x y z α β γ T = f(q1, q2, , q n) =T (1)

For manipulator path planning, the inverse kinematic solution T−1gives the joint angles q

re-quired to reach the specified tool’s position In general this solution is non-unique (Barrientos

et al.; 1997) A generic mathematical representation could be:

q=

q1 q2 q n T

=g(x, y, z, α, β, γ) =T−1 (2)

2.2 Robot Dynamics Model

In the absence of friction or other disturbances, the dynamics of a serial n-link rigid robot

manipulator can be written as (Kelly and Santibáñez; 2003)

M(q) ¨q+C(q, ˙q) ˙q+g(q) =τ (3)where:

M(q)n × n symmetric positive definite manipulator inertia matrix

q n ×1 vector of joint displacements

C(q) n ×1 vector of centripetal and Coriolis torques

g(q) n ×1 vector of gravitational torques

τ n ×1 vector of applied joint torques

2.3 Camera Model

According to Fig 1 we consider a vision system mounted on the robot tool, with coordinateframe, ΣC, which moves in the space< X R , Y R , Z R >of the robot coordinate frame ΣR Theorigin of the camera coordinate frame (tool frame) with respect to the robot coordinate frame

is represented by the vector P R

Corgwith coordinates p R

xc p R

yc p R zc

T

∈ 3in the robot reference system ΣR

2.3.1 Camera Model for Visual-Based 2D Control

The image acquired by the camera supplies a two-dimensional (2D) array of brightness valuesfrom a three-dimensional (3D) scene This image may undergo various types of computerprocessing to enhance image properties and extract image features In this paper we work

with a known spherical object with radius r o The centre of gravity is used as object features(state) for the 2D control We assume that the image features are the projection into the 2Dimage plane of the 3D characteristics of the scene

The object moves in a plane parallel to the plane< Y C , Z C >of the camera coordinate frame

ΣC The camera’s optical axis coincides with X Caxis

A perspective projection with a focal λ is assumed. The point pC

o with coordinates

p C xo p C yo p C zo T in the camera frame projects onto a point (u, v) (pixel) on the imageplane

Fig 1 Camera-in-hand robotic system

(Chaumette and Hutchinson; 2007), but we present the robotic and vision systems modeled

for small variations about the operating point for position control and in these conditions the

stability of the whole system are balanced

A particular study is made using an ASEA IRB6 robot manipulator which has mechanically

decoupled wrist This allows keeping the orientation of the camera’s optical axis while the

arm is moving

In order to validate the proposed control system, a general stability analysis is presented and

an analysis of the step response in a regulator type system is made The disturbance is

inter-preted as initial conditions To illustrate the proposed controller, the control system stability

and its performance, both simulation results as well as experimental results using the ASEA

IRB6 robot manipulator are presented Our experimental results confirm the expected step

response in the image plane, with good time performance and zero steady-state error

2 Robotic System Model

As shown in Fig 1, the robotic system considered has a robot manipulator with a camera in its

hand The basic mathematical description of this system consists of the robot and the camera

model

2.1 Robot Kinematics Model

The kinematics of a manipulator gives the relationship between the joint positions q and the

corresponding tool translational(x, y, z) and angular position(α, β, γ) For an n-axis rigid

link manipulator, the forward kinematic solution, T, could be computed for any manipulator,

irrespective of the number of joints or kinematic structure (Barrientos et al.; 1997) A genericmathematical representation could be:

x y z α β γ T= f(q1, q2, , q n) =T (1)

For manipulator path planning, the inverse kinematic solution T−1gives the joint angles q

re-quired to reach the specified tool’s position In general this solution is non-unique (Barrientos

et al.; 1997) A generic mathematical representation could be:

q=

q1 q2 q n T

=g(x, y, z, α, β, γ) =T−1 (2)

2.2 Robot Dynamics Model

In the absence of friction or other disturbances, the dynamics of a serial n-link rigid robot

manipulator can be written as (Kelly and Santibáñez; 2003)

M(q) ¨q+C(q, ˙q) ˙q+g(q) =τ (3)where:

M(q)n × n symmetric positive definite manipulator inertia matrix

q n ×1 vector of joint displacements

C(q) n ×1 vector of centripetal and Coriolis torques

g(q) n ×1 vector of gravitational torques

τ n ×1 vector of applied joint torques

2.3 Camera Model

According to Fig 1 we consider a vision system mounted on the robot tool, with coordinateframe, ΣC, which moves in the space< X R , Y R , Z R >of the robot coordinate frame ΣR Theorigin of the camera coordinate frame (tool frame) with respect to the robot coordinate frame

is represented by the vector P R

Corgwith coordinates p R

xc p R

yc p R zc

T

∈ 3in the robot reference system ΣR

2.3.1 Camera Model for Visual-Based 2D Control

The image acquired by the camera supplies a two-dimensional (2D) array of brightness valuesfrom a three-dimensional (3D) scene This image may undergo various types of computerprocessing to enhance image properties and extract image features In this paper we work

with a known spherical object with radius r o The centre of gravity is used as object features(state) for the 2D control We assume that the image features are the projection into the 2Dimage plane of the 3D characteristics of the scene

The object moves in a plane parallel to the plane< Y C , Z C >of the camera coordinate frame

ΣC The camera’s optical axis coincides with X Caxis

A perspective projection with a focal λ is assumed. The point pC

o with coordinates

p C xo p C yo p C zo T in the camera frame projects onto a point (u, v) (pixel) on the imageplane

Let pC

o be the position of the object’s centre of gravity According to the perspective projection

(Hutchinson et al.; 1996), we have

ξ=

u v

=− α λ

p C xo

 p C yo

p C zo

(4)

where α is the scaling factor in pixels per meter due to the camera sampling, λ is the focal

length of the camera lens This model is also called the imaging model (Kelly et al.; 2000)

The object distance p C

xo along the camera’s optical axis X Cis constant

The orientation of the camera frame with respect to the robot frame in the plane< Y R , Z R >

Following the configuration of Fig 1, and taking into account equation (5), it is possible to

obtain the y and z components of vector p C

o as:

 p C yo

p C zo

=RC R(ψ)T  p R

yo

p R zo

−  p R

yc

p R zc

−  p R yc

p R zc

(6)

Where h=α p λ C xo

2.3.2 Camera Model for Visual-Based 3D Control

In the 3D control the object moves in the space< X R , Y R , Z R >of the robot coordinate frame

ΣR ; and the camera’s optical axis coincide with the Z Caxis of the camera’s coordinate frame

ΣC We continue work with a known spherical object with radius r o; and in this case the object

features (state) are the centre of gravity and image radio

According to this consideration Equation (4) becomes:

ξ=

 u v

=− α λ

p C zo

 p C xo

p C yo

(7)

In order to estimate the object’s distance p C

zo along the camera’s optical axis Z Cit is possible touse a well-known object size and the corresponding apparent size in the image plane (Corke;

1996) Our object is a sphere with radius r o and the apparent image radius is r Following



=− α λ

p C zo

Let(φ, θ, ψ)being the given set of Euler angles, the following rotations: Frame rotation by the

angle φ about axis Z, frame rotation by the angle θ about axis Y and frame rotation by the

angle ψ about axis Z 

In this case the rotation matrix is:

RR

C(φ, θ, ψ) =



 CφCθCψ − SφSψ − CφCθSψ − SφCψ CφSθ SφCθCψ+CφSψ − SφCθSψ+CφCψ SφSθ



Following the configuration of Fig 1, and taking into account Equation (10), it is possible to

obtain the components of vector pC

p C zo

2.3.3 Linear Camera Model

According to Fig 1, the axes Z R and Z Care parallel and, for simplicity in the analysis, they

are taken with the same direction In this case, the Euler’s angles are φ=0, θ=0 and ψ varies

according the rotation of the robot’s base In these conditions Equation (10) becomes:





δ p C xo

δp C yo

δp C zo

δ p R yo

δp R zo

δp R zc

δξ =



 δu δv δr



=− α λ

p C zo



 δ p C

xo

δ p C yo

− r o δp C zo

p C zo



and finally, according to (13) and (14) we obtain:

Let pC

o be the position of the object’s centre of gravity According to the perspective projection

(Hutchinson et al.; 1996), we have

ξ=

u v

=− α λ

p C xo

 p C yo

p C zo

(4)

where α is the scaling factor in pixels per meter due to the camera sampling, λ is the focal

length of the camera lens This model is also called the imaging model (Kelly et al.; 2000)

The object distance p C

xo along the camera’s optical axis X Cis constant

The orientation of the camera frame with respect to the robot frame in the plane< Y R , Z R >

Following the configuration of Fig 1, and taking into account equation (5), it is possible to

obtain the y and z components of vector p C

o as:

 p C yo

p C zo

=RC R(ψ)T  p R

yo

p R zo

−  p R

yc

p R zc

−  p R yc

p R zc

(6)

Where h=α p λ C xo

2.3.2 Camera Model for Visual-Based 3D Control

In the 3D control the object moves in the space< X R , Y R , Z R >of the robot coordinate frame

ΣR ; and the camera’s optical axis coincide with the Z Caxis of the camera’s coordinate frame

ΣC We continue work with a known spherical object with radius r o; and in this case the object

features (state) are the centre of gravity and image radio

According to this consideration Equation (4) becomes:

ξ=

 u v

=− α λ

p C zo

 p C xo

p C yo

(7)

In order to estimate the object’s distance p C

zo along the camera’s optical axis Z Cit is possible touse a well-known object size and the corresponding apparent size in the image plane (Corke;

1996) Our object is a sphere with radius r o and the apparent image radius is r Following



=− α λ

p C zo

Let(φ, θ, ψ)being the given set of Euler angles, the following rotations: Frame rotation by the

angle φ about axis Z, frame rotation by the angle θ about axis Y and frame rotation by the

angle ψ about axis Z 

In this case the rotation matrix is:

RR

C(φ, θ, ψ) =



 CφCθCψ − SφSψ − CφCθSψ − SφCψ CφSθ SφCθCψ+CφSψ − SφCθSψ+CφCψ SφSθ



Following the configuration of Fig 1, and taking into account Equation (10), it is possible to

obtain the components of vector pC

p C zo

2.3.3 Linear Camera Model

According to Fig 1, the axes Z R and Z Care parallel and, for simplicity in the analysis, they

are taken with the same direction In this case, the Euler’s angles are φ=0, θ=0 and ψ varies

according the rotation of the robot’s base In these conditions Equation (10) becomes:



 δp C

xo

δp C yo

δp C zo

δp R zo

δ p R yc

δp R zc

δξ =



 δu δv δr



=− α λ

p C zo



 δp C

xo

δp C yo

− r o δp C zo

p C zo



and finally, according to (13) and (14) we obtain:

δ p R zo

δp R zc

Recall that, our aim is moving the manipulator maintaining the image of the tracked object

(its centre of gravity) coincident to the centre of the image plane, for the 2D control; and the

image of the tracked object (its centre of gravity) coincident to the centre of the image plane

with a given image radius, for the 3D control

The control problem is formulated as the design of a controller which computes a control

signal ∆ corresponding to the movement of the robot’s arm in such a way that the actual

image object state reaches the desired state

3.1 2D Control Problem Formulation

For 2D control the desired state, u d v d T, is the centre of gravity of the object’s image

The state error being defined as:



which could be calculated at every measurement time and used to move the robot in a

direc-tion allowing its decrease Therefore, the control aims at ensuring that

provided that the initial feature error ˜ξ(0)is sufficiently small

We make the following assumptions for the 2D control problem:

A1.0 The object is static.

A1.1 There exists a robot joint configuration qd for which ξ d=ξ(qd)

A1.2 ψ is the mechanical angle between Σ Cand ΣR, and can take different values but will be

constant in each experiment

A1.3 The axesX R and X C, Fig 1, are parallels

A1.4 The distancep C

xofrom the camera to the object is constant

Assumption A1.0, ensures that only the control problem is evaluated Assumption A1.1,

en-sures that the control problem is solvable Assumption A1.2, stability condition, will be

dif-ferent for each value of ψ Assumption A1.3 maintains the condition of equation (5).

Fig 2 Control scheme with visual feedback

3.2 3D Control Problem Formulation

In the case of 3D control the desired state object’s image centre of gravity and radius is sented by u d v d r d T The state error is defined as



The control aims at ensuring that

We make the following assumptions the 3D control problem:

A2.0 The object is static for position regulation.

A2.1 There exists a robot joint configuration qd for which ξ 

d=ξ (qd)

A2.2 The axesZ R and Z C, Fig 1, are parallel

A2.3 The initial feature error ˜ξ (0)is sufficiently small

A2.4 The object moves with low velocity in following a simple trajectory.

Assumption A2.0, ensures that the regulation problem is evaluated Assumption A2.1, sures that the control problem is solvable Assumption A2.2, conditions for Euler angles Assumption A2.3 makes possible the linear analysis about the operating point Assumption A2.4 conditions of trajectory following.

en-3.3 Controller With Visual Feedback

For our control problem formulation, the state vector of the object can only be measured

through the camera, as such, a direct knowledge of the desired joint position qdis not able Nevertheless, the desired joints position can be obtained as a result of the estimatedcontrol signal ∆ and the solution of the kinematics problems

δ p R zo

δ p R zc

Recall that, our aim is moving the manipulator maintaining the image of the tracked object

(its centre of gravity) coincident to the centre of the image plane, for the 2D control; and the

image of the tracked object (its centre of gravity) coincident to the centre of the image plane

with a given image radius, for the 3D control

The control problem is formulated as the design of a controller which computes a control

signal ∆ corresponding to the movement of the robot’s arm in such a way that the actual

image object state reaches the desired state

3.1 2D Control Problem Formulation

For 2D control the desired state, u d v d T, is the centre of gravity of the object’s image

The state error being defined as:



which could be calculated at every measurement time and used to move the robot in a

direc-tion allowing its decrease Therefore, the control aims at ensuring that

provided that the initial feature error ˜ξ(0)is sufficiently small

We make the following assumptions for the 2D control problem:

A1.0 The object is static.

A1.1 There exists a robot joint configuration qd for which ξ d=ξ(qd)

A1.2 ψ is the mechanical angle between Σ Cand ΣR, and can take different values but will be

constant in each experiment

A1.3 The axesX R and X C, Fig 1, are parallels

A1.4 The distancep C

xofrom the camera to the object is constant

Assumption A1.0, ensures that only the control problem is evaluated Assumption A1.1,

en-sures that the control problem is solvable Assumption A1.2, stability condition, will be

dif-ferent for each value of ψ Assumption A1.3 maintains the condition of equation (5).

Fig 2 Control scheme with visual feedback

3.2 3D Control Problem Formulation

In the case of 3D control the desired state object’s image centre of gravity and radius is sented by u d v d r d T The state error is defined as



The control aims at ensuring that

We make the following assumptions the 3D control problem:

A2.0 The object is static for position regulation.

A2.1 There exists a robot joint configuration qd for which ξ 

d=ξ (qd)

A2.2 The axesZ R and Z C, Fig 1, are parallel

A2.3 The initial feature error ˜ξ (0)is sufficiently small

A2.4 The object moves with low velocity in following a simple trajectory.

Assumption A2.0, ensures that the regulation problem is evaluated Assumption A2.1, sures that the control problem is solvable Assumption A2.2, conditions for Euler angles Assumption A2.3 makes possible the linear analysis about the operating point Assumption A2.4 conditions of trajectory following.

en-3.3 Controller With Visual Feedback

For our control problem formulation, the state vector of the object can only be measured

through the camera, as such, a direct knowledge of the desired joint position qdis not able Nevertheless, the desired joints position can be obtained as a result of the estimatedcontrol signal ∆ and the solution of the kinematics problems

avail-The implemented closed-loop block diagram can be described as shown in Fig 2 avail-The control

system has two loops in cascade, the internal loop solving the robotsŠ joint control, and the

external loop implementing a dynamic look and move visual controller.

The inner control loop has an open control architecture; in this architecture it is possible to

implement any type of controller One possibility is to use a non-linear controller in the state

variables, called torque-calculated (Kelly and Santibáñez; 2003)having the following control

equation:

τ=M(q)[ q¨d+Kvi ˙˜q+Kpi˜q] + C(q, ˙q) ˙q+g(q) Where Kpi ∈  n×nand Kvi ∈  n×nare the symmetric positive-definite matrices and ˜q =

q−qd Kelly (Kelly and Santibáñez; 2003) demonstrated that with this configuration the

system behaves in a closed loop as a linear multivariable system, decoupled for each robot’s

joint, suggesting that the matrices could be specified as:

Kpi=diag { ω2, , ω n2}

Kvi=diag { ω1, , ω n }

In this way each joint behaves as a critically damping second order linear system with

band-width ω i The bandwidth ω i determines the speed of response of each joint In such way

the dynamic effect of the internal loop could be independent with regard to the external loop,

being, according to (Hernández et al.; 2008), under the conditions that:

q(t) =qd(t) ∀ t >0 (17)Nevertheless, the vision-based control systems are fully sampled data systems The feed-

back sensor has some dynamic characteristics such as: transport delay of pixel camera, image

processing algorithms, communication between the vision system and control computer, etc

Åström (Astrom and Wittenmark; 1990) established that the sampling rate of digital control

systems should be between 10 and 30 time the desired closed loop bandwidth For the case of

a 20Hz vision system the close loop bandwidth should be between 0.66 to 2 Hz

With these conditions for the internal and external loop it is possible to make a design in order

to avoid the dynamic effect of the internal loop in relation with the dynamics of the external

loop (Lange and Hirzinger; 2003), a complete analysis of this topic can be found in (Hernández

et al.; 2008)

But, if we analyze the control problem in the field of digital control systems, other dynamic

representation of the set robot vision system could be as one or two delay units for the vision

set (Corke; 1996) or for the robot (Bonfe et al.; 2002) Using this consideration we modify

Equation (17) as,

q(k) =qd(k −1) ∀ k >0 (18)

In this chapter, we consider a simpler approach, which consists of directly using the image

feature vectors ˜ξ for 2D control or ˜ξ for 3D control, being the difference between the centre

of the image plane ξ d or ξ 

d and the centre of gravity of the object in the image space ξ or ξ ;

and for 3D control the difference between the desired radius in the image plane r d and the

actual radius of the object in the image plane r (image coordinate frame) This error depends on

the object’s absolute position, in the task space pR

o, and the camera’s centre position, pR

Corg,according to Equations (6) 2D or (16) 3D

Fig 3 2D Vision-based simplified control scheme

4 2D Vision-based Control Stability Analysis

A very simple I controller can be used in this control scheme (Hernández et al.; 2008), for that

case the control law can be given by:

As in Sim’s work (Sim et al.; 2002), ∆ can be interpreted as the coordinates increment in the

world space as a result of the image feature error ˜ξ Solving the inverse kinematics problem

T −1it is possible to obtain qd The proposed system works as a regulator system, because ξ d

is constant and can be set=0

Taking into account Fig 2, obtaining the discrete equivalence of the controller of Equation(19) and according to Equations (6) and (18); a simplified diagram can be obtained as shown

T

as the system’s initial

conditions, the closed loop transfer function, taking a sampling period of 50ms, can be written

as:

0.05KIK

(z2− z)[U(z)−Y(z)] =Y(z) (21)where

Y(z) =ξ(z)

Solving and taking the inverse Z transform we obtain:

y(k+2)−y(k+1) =−0.05KIKy(k) +0.05KIKu(k) (22)

The implemented closed-loop block diagram can be described as shown in Fig 2 The control

system has two loops in cascade, the internal loop solving the robotsŠ joint control, and the

external loop implementing a dynamic look and move visual controller.

The inner control loop has an open control architecture; in this architecture it is possible to

implement any type of controller One possibility is to use a non-linear controller in the state

variables, called torque-calculated (Kelly and Santibáñez; 2003)having the following control

equation:

τ=M(q)[ q¨d+Kvi ˙˜q+Kpi˜q] + C(q, ˙q) ˙q+g(q) Where Kpi ∈  n×nand Kvi ∈  n×nare the symmetric positive-definite matrices and ˜q =

q−qd Kelly (Kelly and Santibáñez; 2003) demonstrated that with this configuration the

system behaves in a closed loop as a linear multivariable system, decoupled for each robot’s

joint, suggesting that the matrices could be specified as:

Kpi=diag { ω2, , ω n2}

Kvi=diag { ω1, , ω n }

In this way each joint behaves as a critically damping second order linear system with

band-width ω i The bandwidth ω idetermines the speed of response of each joint In such way

the dynamic effect of the internal loop could be independent with regard to the external loop,

being, according to (Hernández et al.; 2008), under the conditions that:

q(t) =qd(t) ∀ t >0 (17)Nevertheless, the vision-based control systems are fully sampled data systems The feed-

back sensor has some dynamic characteristics such as: transport delay of pixel camera, image

processing algorithms, communication between the vision system and control computer, etc

Åström (Astrom and Wittenmark; 1990) established that the sampling rate of digital control

systems should be between 10 and 30 time the desired closed loop bandwidth For the case of

a 20Hz vision system the close loop bandwidth should be between 0.66 to 2 Hz

With these conditions for the internal and external loop it is possible to make a design in order

to avoid the dynamic effect of the internal loop in relation with the dynamics of the external

loop (Lange and Hirzinger; 2003), a complete analysis of this topic can be found in (Hernández

et al.; 2008)

But, if we analyze the control problem in the field of digital control systems, other dynamic

representation of the set robot vision system could be as one or two delay units for the vision

set (Corke; 1996) or for the robot (Bonfe et al.; 2002) Using this consideration we modify

Equation (17) as,

q(k) =qd(k −1) ∀ k >0 (18)

In this chapter, we consider a simpler approach, which consists of directly using the image

feature vectors ˜ξ for 2D control or ˜ξ for 3D control, being the difference between the centre

of the image plane ξ d or ξ 

d and the centre of gravity of the object in the image space ξ or ξ ;

and for 3D control the difference between the desired radius in the image plane r d and the

actual radius of the object in the image plane r (image coordinate frame) This error depends on

the object’s absolute position, in the task space pR

o, and the camera’s centre position, pR

Corg,according to Equations (6) 2D or (16) 3D

Fig 3 2D Vision-based simplified control scheme

4 2D Vision-based Control Stability Analysis

A very simple I controller can be used in this control scheme (Hernández et al.; 2008), for that

case the control law can be given by:

As in Sim’s work (Sim et al.; 2002), ∆ can be interpreted as the coordinates increment in the

world space as a result of the image feature error ˜ξ Solving the inverse kinematics problem

T −1it is possible to obtain qd The proposed system works as a regulator system, because ξ d

is constant and can be set=0

Taking into account Fig 2, obtaining the discrete equivalence of the controller of Equation(19) and according to Equations (6) and (18); a simplified diagram can be obtained as shown

T

as the system’s initial

conditions, the closed loop transfer function, taking a sampling period of 50ms, can be written

as:

0.05KIK

(z2− z)[U(z)−Y(z)] =Y(z) (21)where

Y(z) =ξ(z)

Solving and taking the inverse Z transform we obtain:

y(k+2)−y(k+1) =−0.05KIKy(k) +0.05KIKu(k) (22)

Fig 4 Root-locus of axis model for: a) I controller with closed poles for gain hK I i =10 b) PI

controller with Closed poles for gain hK I i =30

Taking into account that the aim is moving the manipulator maintaining the image of the

tracked object (its centre of gravity) coincident to the centre of the image plane, we can make

u=0 and v=0, then u=0 and (22) becomes

y(k+2) =y(k+1)−0.15KIKy(k) (23)The Equations (23) can be represented in the state space as

In this work a robot manipulator ASEA IRB6 with camera in hand is used as case study This

type of robot has the wrist mechanically decoupled from the arm’s movements, this allows to

maintain the orientation of the camera It is possible to establish ψ constant and for simplicity

we set ψ=0

With this consideration and according to Equation (4) Equation (24) can be transformed into

two decoupled control systems, one in each axis The representation of these systems in the

space state is:

The system’s stability is determined by the roots of the characteristic polynomial of matrix Φ.

In these conditions it is easy to design the regulator for each axis

A better illustration is shown by the root-locus diagram of the systems (25) or (26), given

in Fig 4 a), where instability is clearly indicated as the loop gain increases If we use a PI controller, including a zero in z =−0.2, the diagram of Fig 4 a) is modified as is shown inFig 4 b) In this case the stability condition of the system has been increased

With the PI controller Equations (25) and (26) are modifed as:

5 3D Vision-based Control Stability Analysis

Similar to the 2D vision-based control analysis, we use a very simple I controller in the control

scheme, for that case the control law is given by:

In this case too, ∆can be interpreted as the coordinates increment in the world space as a

result of the image feature error ˜ξ  Solving the inverse kinematics problem T −1it is possible to

obtain qd An analysis of this control scheme in continuous time can be found in (Hernández

et al.; 2008a), where also the control performance in following simple trajectories in 3D ispresented

The system works as regulator, because ξ 

d is constant and can be set= 0 In these tions, following the analysis of Section 4, using Equations (15) and (29) the system’s output

condi- u(k) v(k) r(k) T, can be obtained as decoupled equations by axis, as:

T

as system initial conditions

With ψ = 0 as operating point, for the three previous Equations, according Equation (16),only (33) is enabled Its stability analysis can be done in a Root-locus similar to Fig 4 For the

analysis along the axes v and u Equation (16) can be simplified as: