Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 15 pps

For any given robot configuration with multiple manipulators, links, branches, etc., it is possible to enumerate potential sensor placements, divide the system into subsystems at the sen

When there are two unknown forces (such as in Fig 7(c)) the forces and torques cannot be calculated directly This situation can often be solved by starting at several points in the chain and propagating the known forces and moments to a common point In other cases additional information (such as provided by an additional force/torque sensor) is needed to permit a solution When all of the links in the system have been visited it is possible to determine if the given set of sensors is sufficient or if additional sensors are required

2.3 Analysis of Canonical Elements

The above analysis can be applied case by case to the canonical elements in Fig 3 First consider the chain with known loads (Fig 3(a)) Starting with the link on the far left, it is possible to find the actuator torques on the first joint Continuing with the links from left to right it is possible to find forces and torques on all joints in this system Hence there are enough sensors to completely identify all actuation efforts for this case

The canonical element chain with one unknown load also has enough sensors, but it is necessary to work inward from both ends of the chain simultaneously so that the single unknown load at the middle link can be determined However, for any chain that has more than one unknown load (as in Fig 3(c)) all actuator efforts cannot be determined without adding more sensors

Loops can be resolved into two chains joined by two branching links Loops are analysed by starting with a link that has only known applied loads and propagating the loads in both directions around the loop until the chain rejoins It can be shown that there are not enough sensors to determine all actuation efforts for any of the three canonical elements with loops However, inserting a sensor in a loop converts this problem into the case of the chain with known loads (Fig 3(a)) To summarize, for all the canonical elements, only a chain with known loads (Fig 3(a)) and

a chain with one unknown load (Fig 3(b)) have enough sensory information to determine all actuation efforts

Using the analysis of the canonical elements, it is straightforward to apply the results to the original system to determine sensor placement For any given robot configuration with multiple manipulators, links, branches, etc., it is possible to enumerate potential sensor placements, divide the system into subsystems at the sensors, classify each subsystem by its canonical element, eliminate the layouts where there is not enough sensory information, and find the minimal number and placement of sensors for the system

3 System Topologies

Systems such as Fig 2 were studied to determine the torques at each joint and the reaction jet forces The parameters varied are number of manipulators (p = 1, 2), number of links per manipulator (n = 1, 2, many), reaction jets or not (free-flying or free-floating), and payload or not The primary locations for sensor placement are the manipulator wrist and the manipulator base where it joins the spacecraft For most cases it is not necessary to enumerate all the cases where the sensor is placed at any joint in between since they are often equivalent to the cases where there are sensors at the ends of manipulator

3.1 Minimum Sensor Configurations

The sensor placement method presented above was applied to space robots with one and two manipulators A collection of single manipulator cases with and without thrusters is summarized in Fig 8 In all cases with a single manipulator there was adequate sensing Fig 9-12 summarize the results for space robots with two manipulators The cases

in Fig 9 do not have (or are not firing their) thrusters The first row shows possible sensor placements when one sensor is available It can be placed between the manipulator and the spacecraft, at the end effector, between both manipulators and the spacecraft, or at the end of both end-effectors The second row shows placement of two sensors, the third three sensors, and the last row the only configuration with four sensors All cases reduce to the canonical chain elements with at most one unknown load, except for the two loop cases which are crossed out The crossed out cases do not have enough sensing to determine all actuation efforts From the remaining cases which do have enough information it is easy to determine the minimal sensors (one) and their potential locations These cases are outlined in bold Fig 10 shows the same cases as Fig 9, except that the space robots now have thrusters The addition of the unknown thruster loads does not change the results There are still only two cases which do not have enough sensing, and a single force torque sensor is enough to determine actuation

Fig 11 shows robots which have no thrusters but carry a payload grasped by both manipulators creating a closed loop Most of the loops are broken by a sensor so actuation can be determined but there are only two places to put a single sensor to determine actuation Fig 12 shows the robots from Fig 11 with thrusters Once again addition of unknown thruster forces does not significantly change the results

s

Figure 8 Space robot configurations for a single manipulator, no thrusters

Figure 9 Space robot configurations for two manipulators and no thrusters

Figure 10 Space robot configurations for two manipulators and thrusters

Figure 11 Space robot configurations for two manipulators and payload

Figure 12 Space robot configurations for two manipulators, payload, thrusters

be accomplished but control algorithms are beyond the scope of this paper Fig 14 shows the results of the same sensor measurements used

to estimate the applied spacecraft reaction jet forces The actual reaction jet forces do not match the commanded due to nonlinear effects However the method is still able to estimate these forces

−2000

−1000 0 1000 2000

time [sec]

Commanded Torque

Manipulator 1 Joint 1

Estimated Torque (dashed line) and Applied Torque (solid line) virtually the same

time [sec]

F

x [N]

Commanded Spacecraft Force

Estimated Force (dashed line)

Applied Force (solid line)

Maximum Error <5%

Figure 14 Continuous commanded net thruster forces, satellite capture task

5 Conclusions

In space robots, actuator effort sensing is required for precise control However, such sensing adds complexity, weight and cost Hence it is important to minimize the number of sensors used Here it is shown that there are minimum sensor configurations that are able to determine system actuation precisely It was found that a base force torque sensor

are included here to provide sensor redundancy With failure of one sensor this system could still maintain precise control While the above resultsare valid for 3D systems, for clarity 2D cases of a satellite capture taskwere simulated The simulations were done in Matlab for the free-flyingspace robot, firing its thrusters at the same time as the manipulatorend-effectors were tracking the grasp points on the satellite [Boning 2006] Fig 13 shows joint torques for the first manipulator The mani-pulators have very high friction, 50% of their maximum capable tor-que The method’s torque estimate and actual applied torque agree

for each manipulator can provide an estimate for friction in the joints and applied reaction jets A wrist force torque sensor for each manipulator can also be used to estimate joint friction and applied reaction jet forces However, additional sensors are needed for cases when there are closed kinematic loop configurations

The methods shown here can be applied to other situations such as unknown contact forces at the end-effector, unknown payload mass, or payload gripped with pin joints (rather than rigidly grasped) Systems with reaction wheels can be considered with this methodology The approach is useful to study redundant sensor configurations and accommodate sensor failure Current studies are underway to consider the effects of higher order dynamics and sensor noise An experimental validation of the method is expected to be completed shortly (in time for the ARK conference)

Acknowledgment

The support of the Japan Aerospace Exploration Agency (JAXA) is acknowledged by the authors The authors would like to thank Prof Ohkami, Mr Ueno, and Mr Ishijima for their useful comments

References

Boning, P and Dubowsky, S (2006), Identification of Actuation Efforts using

Limited Sensory Information for Space Robots, Proc of the IEEE International Conference on Robotics and Automation, (to be published)

Breedveld, P., Diepenbroek, A., van Lunteren, T (1997), Real-time Simulation of

Friction in a Flexible Space Manipulator, Proceedings of the 8th International Conference on Advanced Robotics, Monterey, CA

Satellite Capturing Strategy using Agile Orbital Servicing Vehicle,

Hyper-OSV, Proc of the IEEE Int Conf on Robotics and Automation, Wash DC

Morel, G., Iagnemma, K., and Dubowsky, S (2000), The Precise Control of Manipulators with High Joint-Friction Using Base Force/Torque Sensing,

Automatica: The Journal of the International Federation of Automatic Control,

Staritz, P.J., Skaff, S., Urmson, C., and Whittaker, W (2001), Skyworker: A Robot for Assembly, Inspection and Maintenance of Large Scale Orbital

Facilities, Proc IEEE Int Conf on Robotics and Automation, Seoul, Korea

Ueno, H., Nishimaki, T., Oda, M., and Inaba, N (2003), Autonomous Cooperative

Robots for Space Structure Assembly and Maintenance, Proc 7th Int Symp

on Artificial Intelligence, Robotics, and Automation in Space, NARA, Japan

Vischer D and Khatib O (1995), Design and Development of High-Performance Wilson, E., Lages, C., and Mah, R (2002), Gyro-based maximum-likelihood

thruster fault detection and identification, Proc of the American Control Conference, Anchorage, AK

Matsumoto, S., Ohkami, Y., Wakabayashi, Y., Oda, M., and Ueno, H (2002),

No 7, vol 36, pp 931-941

Torque-Controlled Joints, IEEE Trans Robotics & Automation, No 4, vol 11

TRANSLATING 3-CCR/3-RCC

PARALLEL MECHANISMS

Massimo Callegari and Matteo-Claudio Palpacelli

Polytechnic University of Marche, Department of Mechanics

Ancona, Italy

{m.callegari, m.palpacelli}@univpm.it

Abstract The paper presents the kinematics of the 3-RCC/3-CCR translating parallel

mechanisms and several machines of such family are described in detail taking into account different possible kinds of actuations They all share good kinematic properties as for instance simple closed-form relations and convex workspace, but differ for overall dimensions of the mobile platform and dynamic behaviour: therefore the concepts have been optimized and compared against common performance indices, to determine the best solutions for selected classes of applications Based on such results, a prototype robot has been finally built

Keywords: Parallel robots, Translating Parallel Machines, Kinematic analysis,

1 Introduction

The kinematics of parallel mechanisms with full spatial mobility is usually very complex and sometimes cannot even be solved in closed form, as for example in the well-known case of the general Gough-Stewart platform (Liu and Fitzgerald, 2003) For this reason it is recently growing the interest of researchers and industry towards minor mobility mechanisms, able to perform elementary motions like pure rotations, pure translations or planar displacements: in this way the complexity of the analytical problem is greatly reduced while the advantages of closed-loop actuation are still preserved, eventually by having two parallel mechanisms mounted in series or cooperating in parallel for the accomplishment of a given task

Many new translating parallel machines (TPMs) have been studied in the last years, including the 3-RPC mechanism by (Callegari and Tarantini, 2003) that showed good kinematic performances, e.g simple equations, high stiffness, convex workspace, and so on However the aforementioned concept presented also some clear weak points, such as a cumbersome moving platform, poor dynamic performances and an

© 2006 Springer Printed in the Netherlands

J Lenarþiþ and B Roth (eds.), Advances in Robot Kinematics, 423 423–432

Conceptual design, Optimization

OF THE

overconstrained structure: therefore the study has been enlarged to the whole family of machines with 3-RCC/3-CCR kinematics (Callegari and Marzetti, 2003) to assess whether other mechanisms of this set show better features than the one previously identified

The paper briefly presents three mechanisms selected among the most

optimization and compares the resulting performances

2

A whole family of mechanisms can be defined by the functional design schematically represented in Fig 1a: a mobile platform is connected to the fixed base by three identical limbs, that are composed by two members coupled by a cylindrical pair; the lower link of each leg is connected to the frame by a revolute joint while the upper one is connected to the platform by a cylindrical pair Such kind of mechanisms are conventionally called 3-RCC to indicate the sequence of the joints in the three (identical) limbs, starting from the fixed frame and moving towards the platform; the 3-CCR architecture, shown in Fig 1b, is simply obtained by kinematic inversion It can be easily seen that the described architecture is characterized by 3 d.o.f.’s in space where in the general case spatial translations are coupled with changes of orientation of the platform Nevertheless for particular geometrical configurations such mechanisms can provide motions of pure translation, i.e iff (i) the axes of

the revolute and outer cylindrical joint of i th limb are parallel to the same unit vector uˆi and (ii) u ˆ ≠i u ˆj for i  j (i,j = 1,2,3)

It must be pointed out that in these cases the two links of each leg do not turn around the cylindrical pair, which could well be substituted by a prismatic joint, giving rise to the already mentioned 3-RPC (or 3-CPR) overconstrained mechanism

Description of the Family of Mechanisms

Figure 1 The 3-RCC (a) and 3-CCR (b) parallel mechanisms

to the possible different actuations and finally performs a kinematic interesting elements of this class, develops their kinematics according

3

In this section three different mechanisms are considered The 3-RCC architecture shown in Fig 1a can be actuated either by driving the base revolute joints or by controlling legs’ extensions: the mobile platform of such concepts turns out to be rather bulky because of the (unavoidable) length of the stroke of the cylindrical pairs mounted on the platform itself This drawback can be possibly avoided by inverting the kinematic structure of the mechanism, Fig 1b: in this case the legs can be actuated

by directly driving the sliders running along the fixed slideways Unfortunately a mere inversion of the mechanism is not possible because the resulting kinematics shown in Fig 1b is singular in its entire workspace; therefore two more concepts are presented, slightly different from first mechanism but both characterized by a 3-CCR architecture

3.1

The configuration of the mechanism is symmetric, Fig 1a, with the axes of the three revolute pairs forming an equilateral triangle on the fixed base; in the same manner, another equilateral triangle is formed on the moving platform by the axes of the cylindrical pairs; moreover, the legs are perpendicular to the joints connecting them with the two bases

In case the machine is driven by controlling legs’ lengths, IPK is characterized by a single configuration of the mechanism while DPK presents 8 different solutions that can be simply evaluated in closed-form (Callegari and Tarantini, 2003) When the unit vectors of the three limbs become linearly dependent, the manipulator gets stuck in a singular configuration: therefore the locus of all the singular points is given by a right cylinder, that however can be moved outside the workspace with a

mechanism can be easily driven by means of rotary motors lumped at the end of the limbs and ball screws, see Fig 2: such design yields very good static properties (e.g a high thrust can be delivered since the legs are loaded only by normal forces) but the resulting dynamic behaviour is poor due to the relevant mass of the mobile platform and the high inertia Better dynamic performances can be obtained by directly powering the base revolute joints, even if the limbs are loaded by bending moments in this case (Callegari and Marzetti, 2003) developed the complete kinematics of such mechanism: it is shown that a (very simple) unique solution exists for both direct and inverse position problem and no translation or rotation singularities are possible at all

Kinematics of Some Mechanisms

3-RCC with Triangular Configuration

proper dimensioning of the machine No constraint (rotation) singu- larities exist apart from the surface of the mentioned cylinder This

of the spinning or tilting masses (Callegari et al., 2006)

3.2

The mere kinematic inversion of the 3-RCC mechanism just described

is shown in Fig 1b: (Callegari and Marzetti, 2003) proved that such

As a matter of fact, in order to give full mobility to the inverted mechanism it is necessary that the direction of the axes of the three pairs connecting the limbs to the ground are linearly independent

th

For instance, in the tetrahedral configuration shown in Fig 3a such axes stem from the origin of the fixed frame O(x,ˆy,ˆzˆ) and are tilted by α radians with respect to the horizontal plane If a frame P(u,ˆv,ˆwˆ), parallel

to O(x,ˆyˆ ,zˆ), is attached to the mobile platform at the intersection of the

Figure 2 Sketch of the 3-RCC translational platform (actuation on legs’ length)

3-CCR with Tetrahedral Configuration

limb Figure 3 Sketch of the tetrahedral 3-CCR architecture and loop-closure for i

mechanism is useless since all the points of its workspace are singular

three revolute joints, the vector p=OP can be taken to specify the

position of the moving platform

In order to simplify the study of the mechanism, other frames can be

easily defined for each link as shown in Fig 3b: e.g the frame

A , attached at of ith limb, is obtained by starting at the global

frame O(x,ˆy,ˆzˆ), then it is rotated by ϕi around the (current) z i axis and

then by α around the (current) y i axis, finally a translation a i along the

direction of the (current) x i axis is performed, to allow for the variable

sliding of the cylindrical pair Moreover, the articular coordinates of the 3

joints are defined as follows: a i and ϑi for the first cylindrical pair, d i and

βi for the second one and γi for the revolute joint It is also noted that the

configuration of Fig 3 is characterized by maximum symmetry, therefore

i d t a

i i i i i

z y x

i i

i i

i i

c

s d s s

s c

c t a

p p p

c s s s c

c s

s c s c c

ϑ

ϑβ

ϑ

βϑβα

αϕαϕ

ϕϕ

ααϕα

0

0

The mobility analysis developed in the Appendix shows that

unfortunately such architecture is liable of constraints singularities

inside its entire workspace; therefore, in order to prevent it rotating, it is

necessary to turn to an overconstrained architecture by replacing limbs’

internal cylindrical pairs with a prismatic pairs, resulting with a 3-CPR

kinematics The following analysis will be performed with reference to

this case, where Eq 2 still holds with βi= 0

Inverse position kinematics relations provide the actuated variables as

functions of platform s position p:

i z i y i x

The same Eq 3, written for i=1,2,3, is used to solve the direct position

kinematics and find the values of p x,p y,p z as functions of a i The direct

derivation of Eq 1 is the base for the velocity analysis:

i i i i i i

By dot-multiplying both sides of Eq 4 by the unit vector aˆi and

collecting the 3 relations for i=1,2,3 in a single expression in matrix form,

a

vp= 

J , the expressions of the Jacobian matrix is obtained:

’

ααϕαϕ

s c s c c

s c s c c

s c s c c J

T T T

3 3

2 2

1 1

3 2 1ˆˆˆ

a a

a

(5)

The constant Jacobian shows that this mechanisms belongs to the

class of the so-called Cartesian parallel robots (Kim and Tsai, 2002) It is

noted that the maximum value of det =( )J c2αsα, granting an “optimal”

manipulability, is obtained for α≈ 35 26 ° which corresponds to aligning

the ground cylindrical pairs along a Cartesian frame

3.3

The setting of the joints shown in Fig 4 gives rise to a full Cartesian

kinematics; in fact inverse and direct position kinematics are solved by:

t s p c p a

t p a

3

2 2

1 1

1 1

a p

s

t t c a a c p

t a p

z y x

0

001ˆ

ˆˆ

2 2

1

α

α s c J

T

T T T

a a

a

(7)

showing that J is the identity matrix when the three ground cylindrical

affected by rotation singularities inside the workspace therefore, just as

before, the equivalent 3-CPR mechanism should be used instead

3-CCR with Cartesian Configuration

pairs are orthogonal one to the others (α = 90°) Also this architecture is

Figure 4 Sketch of the Cartesian 3-CCR architecture

4

The kinematic performances of the described machines strongly

depend on the specific geometric parameters of the four architectures:

the machines have been individually optimized and then the resulting

performances have been compared The following non-dimensional

performance index has been used for the optimization:

3 3 2 2 1

where F 1 measures the volume of the workspace, F 2 is proportional to

mobile platform s overall dimensions, F 3 quantifies the dexterity of the

i

condition: β1+β2+β3 =1, the performance index results bounded by 0 ” PI ”1

The measure of workspace volume F 1 is chosen with reference to the

Monte Carlo method proposed by (Stamper, 1997): robot s workspace is

inscribed inside a cube of side 2d max and is discretized in n tot points; then

a numerical procedure determines the number of n in points belonging to

robot s workspace and it is finally defined:

tot

in n

where t max is the maximum dimension of its height and is constrained to

the maximum stroke of the ground sliders Machine s dexterity is

assessed by computing the condition number c of the matrix J T J, with J

Jacobian matrix, as suggested by (Gosselin and Angeles, 1991); by using

the already defined discretization, it is therefore defined:

in

n

tot in tot

n

n d n

c d dW

dW c F

in in

1 23 max 2

3

18

18

1

(11)

The variable parameters of the optimization are the geometric

dimensions of both platforms, limbs lengths and the incidence angle α

between joint axes (or the offset distance e for the fully Cartesian

machine); in order to deal with dimensionless parameters all variables

are divided by maximum limbs’ lenght or by the stroke of ground sliders,

i.e d max = 1 or a max = 1 is imposed according to the specific architecture

If the three functions F i are equally weighted, i.e β1=β2=β3, the

Optimization of the Families

therefore according to what has been proposed by (Carretero et al., 2000),

mechanism and β , i=1,2,3 are the weights of the 3 functions; since

’

the two CPR concepts yield the best performances also when the two

3-RCC architectures present a relatively high PI in spite of a virtually null

workspace In fact the optimization process drove geometry variations of the 3-RCC concepts towards very small values of mobile platforms sides, therefore constraining the displacements of its cylindrical pairs

(Tetrahedral)

CPR (Cartesian)

Figure 5 Optimized workspace of the 4 machines: 3-RCC with rotary (a) or

linear (b) actuation, 3-CPR with tetrahedral (c) or Cartesian (d) configuration

Table 1 Comparison of robots performances (optimization with equal weights)

.

5 Conclusions

The paper has presented the kinematic properties of the translating platforms characterized by the 3-RCC architecture: it has been shown that also its 3-CCR kinematic inversion can be considered but, due to rotation singularities, the rotation of the inner cylindrical joint must be avoided by turning to the 3-CPR overconstrained architecture After having optimized the mechanisms, it has been shown that the two 3-CPR concepts present quite better performances than the others: as a matter

of fact they are both Cartesian parallel machines, being slightly different only due to the disposition of joint axes in the space and present a (comparably) wide workspace characterized by very high dexterity Moreover by still keeping the same geometry of the 3-CCR tetrahedron design of Fig 3, but with a different disposition of the pairs, i.e the 3-CPU architecture shown in Fig 6a, it is possible to get rid of all singularities and obtain a non-overconstrained machine that still retains the good feature of the original concept (as a matter of fact the same kinematics equations at all) A prototype robot has been built with this the first experimentations are still under way

structure, as described in (Callegari et al., 2005), and is shown in Fig 6b:

Figure 6 Concept (a) and prototype (b) of the 3-CPU translation robot

Control of the 3-RCC Translational Platform, submitted to Mechatronics

Conf Robotics and Automation, Barcelona, April 18-22, pp 4031-4036

parallel manipulator assembled for motions of pure translation, Proc Intl Mechanisms, Proc Intl Workshop on Robotics in Alpe-Adria-Danube Region,

Callegari, M., Tarantini, M (2003), Kinematic Analysis of a Novel Translational

Platform, ASME Journal of Mechanical Design, vol 125, pp 308-315

Carretero, J.A., Podhorodesky, R.P., Nahon, M.A., Gosselin, C.M (2000), A

Global Performance Index for the Kinematic Optimization of Robotic

Manipulators, ASME Journal of Mechanical Design, vol 122, pp 17-24

Gosselin, C., Angeles, J (1991), A Global Performance Index for the Kinematic

Optimization of Robotic Manipulators, ASME Journal of Mechanical Design,

vol 113, pp 220-226

Kim, H.S., Tsai, L-W (2002), Evaluation of a Cartesian parallel manipulator In

Advances in Robot Kinematics: Theory and Applications, Lenarcic and Thomas

Liu, K., Fitzgerald, J.M (2003), Kinematic Analysis of a Stewart Platform

Manipulator, IEEE Trans Industrial Electronics, vol 40, pp 282-293

Stamper, R.E (1997), A Three Degree of Freedom Parallel Manipulator with

Only Translational Degrees of Freedom, Ph.D Thesis, University of Maryland,

College Park, Maryland

Appendix Mobility analysis of the tetrahedral 3-CCR

The angular velocity ω of mobile platform can be expressed in function

of i th limb’s articular coordinates as follows, see Fig 3b:

i i i i i

Ȧ=ϑ⋅ˆ +β⋅ˆ +γ⋅ˆ (12)

By dot-multiplying Eq 12 by dˆi, it is obtained:

i T

=

⋅+

⋅+

i i i i z y

i x

i c p s c p s p a t s z II

If platform’s motors are held still (a i =0), due to the absence of

translation singularities is also: px= py= pz =0 and therefore from Eq

14:

i i

i s

Initial mounting conditions require βi =0, therefore it is evident that a

finite angular velocity Ȧ can be initiated: however, since in case of pure

translations the legs do not rotate around their own axes, the problem

can be overcome by the introduction of prismatic pairs in place of the

inner cylindrical joints (3-CPR architecture)

Eds., Springer, pp 21-28