Analysis and solution of the trajectory planning problem for reconfigurable two-arm robots

Preprints 01 the Fourth IFAC Symposium on Robot Control September 19-21, 1994, Capri, Italy

ANALYSIS AND SOLUTION OF THE TRAJECTORY PLANNING PROBLEM FOR RECONFIGURABLE TWOARM ROBOTS V. TAUSSAC·, E. TABARAH·; E. DOMBRE· and B. BENHABIB·· ·Laboratoire d'Informatique de Robotique et de Microelectronique de Montpel/ier, UM II - CNRS 9228, 161 rue Ada, 34392 Montpellier Ceder 5, France ··Computer Integrated Manufacturing Laboratory, Department of Mechanical Engineering, Univer$ity of Toronto , 5 King '$ Col/ege Road, Toronto, Ontario, Canada, M5S 1A4

Abstract. Two-arm systems configured with modular robots are used to perform tasks in coordinated manner. The synthesis problem, in this context, would address issues such as , distribution of degrees of freedom (dof) between the two-arms , determination of geometries , and relative placement of their bases. This paper addresses the first task in the process of solving the synthesis problem, namely the development of an "analysis" module used to evaluate and compare different robot configurations for a desired operation. An optim.ization procedure, based on a trajectory-approximation, is presented herein for determining best continuous path end-effector trajectories for planar systems. Key Words. Modular robots; two-arm robots ; optimal coordination; workspace ; trajectory resolution

1. INTRODUCTION Reconfigurable modular robots are defined as systems that consist of standard modular devices , such as actuators, links , and connectors , utilized to configure "special- purpose" robotic arms (Cohen et al. , 1992). The industrial application of such systems would relieve the heavy financial burden of acquiring sophisticated and generalpurpose robots. Modular robots, by definition , offer the possibility of easily and inexpensively reconfiguring the workcell to suit new tasks. There exists an abundance of literature on the subject of modular robots , which has started over a decade ago, with an aim at industrial as well as space applications. Reference (Tesar and Butler, 1989) provides an excellent review of the pertinent literature . There also exists a wealth of literature dealing with cooperative two-robot systems. The emphasis has usually been on the use of existing robots in order to form closed-chain systems (Zheng and Luh, 1988 ; Craigan and Akin , 1989) . Such systems have been shown to offer a significantly improved load-carrying capability and force /torque output , as well as an improved rigidity (Tarn et al .. 1988). However , a novel method has recently been proposed (Tabarah et al., 1992 ; Jouaneh et aI. , 1990) which advocates the coordination of two robots where, instead of the two robots forming a closed-chain system and operating on a fixed workpiece, the workpiece is gripped by one robot while the tool is mounted on the second robot:

the two robots are coordinated to move so as the tool performs its required motion relative to the workpiece , Fig. 1. The advantages of maneuvering the workpiece , while it is being operated on, include the fact that the trajectory can be made more accessible to the tool , thus facilitating the task and reducing the required workspace , as well as allowing certain complex operations, that would otherwise be infeasible for a fixed workpiece, to be performed in a single run . It has been shown , for instance, that coordinated tworobot systems can perform certain manufacturing operations faster , requiring less power , than using a single robot . These operations include contact operations, such as deburring, which require that the tool maintains contact with the workpiece as it moves , or non-contact operations , such as arc-welding, which require no contact between the tool and the workpiece. The objective of our work is to demonstrate that modular robots can be utilized effectively in forming two-arm systems, where each arm may have a number of degrees of freedom (dof) between one and six (Tabarah et al., 1993). The selection of the type and exact number of dof for the system, for a given task , the selection of the number of dof between the two arms (the dof-distribution problem), as well as the relative placement of their bases is a synthesis problem. We have developped an "analysis" module to be used for the evaluation and comparison of different solutions to this problem. It implements an optimization proce-

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dure for determining best continuous path endeffector trajectories.

3. TRAJECTORY RESOLUTION Trajectory resolution can be classified as either "exact" or "approximate". Exact trajectory resolution requires full mobility for both arms. Once a trajectory has been arbitrarily guessed for arm-1 it could be sampled into K points, where K is very large, as dictated by the robot controller, and defined as functions of time. In turn, the tool-trajectory on the object is also sampled into K points. Since both arms have full mobility, once a point on arm- 1's trajectory is specified, arm-2 can meet this point by bringing the corresponding specific point on the object to arm- 1's trajectory. It is important to note however that, if arm-2 does not have full mobility , the above "rendez- vous" procedure would only work partially. Namely, once a point is chosen on arml's trajectory we would have to solve a pseudosynthesis problem in order to determine the corresponding contact point on the object. Thus, it could not be arbitrarily (and optimally) chosen as in the case of arm- 2 having full mobility. In both cases, the collection of "rendez- vous" points forms the conjugate trajectory for arm- 2. Approximate trajectory resolution does not require full mobility. It is based on the assumption that even when K is very large the obtained conjugate trajectories are never exact. This assumption is further strengthened by the following fact: when we do not have full mobility, even if we satisfy the K positional constraints , due to their unknown placement on the object , we would still have large deviations from desired relative velocities between the tool and the object.

Fig. 1. General configuration of a two-robot coordinated system

2. PROBLEM STATEMENT This paper addresses the coordination problem between two robots performing a common manufacturing-type task, such as welding or deburring , on a workpiece . In order to clarify the trajectory resolution issue, let us consider a twoarm planar system. Arm-1 manipulates a tool that tracks a desired tool- trajectory prescribed relative to a workpiece held and maneuvered by arm-2, Fig. 1. If the two- arm system is a nonredundant one, this trajectory is resolved into a simple pair of new trajectories for the two robots. These new trajectories, termed here as conjugate trajectories, are specified in task space for the two robots' end effectors. In the case where the system has redundant dof, there exists an infinite number of pairs of conjugate trajectories for a given tool trajectory. Thus , the problem at hand can be defined as the optimal determination of conjugate trajectories for two-arm systems with redundant dof.

Based on this assumption, a trajectory approximation procedure, defined in the literature in other contexts , will be proposed here for two-arm robotic systems. The procedure , in its conceptual form, is as follows:

In this paper , we are encouraging the use of reconfigurable modular robots where the two arms may be configured to have less than six dof each. The number of redundant dof, depending on the task , may thus vary between zero and six for a general three-dimensional task. Our analysis shows that the trajectory-resolution problem must then be addressed individually for each of the different numbers of dof- redundancy cases. The different solutions would be based on a generalized workspace-analysis procedure , where the search space for the optimal trajectories (i.e., the system 's operational workspace) is governed by the geometry of the two arms as well as the designated task.

(a) Approximate the tool-trajectory, defined on the object, by K points. (b) Based on a desired relative velocity that the tool has to follow , describe the K locations on the tool- trajectory as a function of time , [.c(t)) i i = 1, I<. (c) Solve the inverse kinematics of both arms individually at [.c)i , where the tool is in contact with the object. It must be noted that in order to solve the inverse kinematics , the locations [.c)i have to be defined in a "world" reference-frame coordinates Fw: (i) If the two arms , forming a closed-chain loop at [.c(t))i , have no redundant dof, then [.c(t))i is uniquely defined with respect to Fw. The inverse kinematics yield unique arm configurations (O l )i for arm- 1 and (02)i for arm-2. (ii) If the two arms have some redundant dof,

The next two sections will discuss this trajectory resolution problem and introduce an optimization procedure for selecting one solution based on certain optimality criteria.

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(d)

(e) (f)

(g)

then [C(t)]i can be located anywhere in the workspace subject to some constraints. Thus, one concludes that there exists infinite arm configurations corresponding to a single relative [C(t)k In other words, the problem at hand is the absolute placement of [C(t)]i with respect to Fw . It is very important to remember that solving this optimization for the [C(t)]i individually would only yield a local optimal. In order to obtain a global optimal, all [C(t)]i have to be optimally determined in a simultaneous matter. Fit cubic splines to arm configurations, de1, 1<. fined by the optimal (Ol)i and (02)i, i As was shown in (Tabarah et al. , 1993) , the time intervals set earlier in step (b) in a normalized fashion can be expanded or shrunk here to satisfy joint velocity and acceleration constraints, thus yielding an optimal overall motion-time for example. Determine both conjugate trajectories for the end effectors of the two arms. Based on the (simulated) actual (relative) tool trajectory calculated in step (e), determine the position , orientation and velocity errors as functions of time, or locations on the desired tool trajectory. If errors determined in step (f) exceed predefined tolerances , the number of points K has to be increased and the optimization procedure returns to step (b).

arm system, the contact point under consideration thus can be placed anywhere on an arc, referred to here as the "iso-pos" curve ("pos" standing for position). Generalizing this case, one notes that every contact point chosen would have a corresponding iso-pos curve. The combination of these iso-pos curves forms the search space for the optimization problem, Fig. 2. The proposed solution

=

Fig. 2. The task

for the optimal placement of [C]i for the two-arm system considered in Fig. 2 is as follows: (i) Determine the iso- pos curves corresponding to [C]i , i = 1, J< contact points. (ii) Consider a straight line defined with respect to Fw by y = ax + b; and, arbitrarily choose a set of (a, b) coefficients so that the straight line intersects all the iso- pos curves. (iii) Determine all intersection points in Fw coordinates and execute the inverse kinematics solution procedure for both arms. (Since both arms are individually non- redundant , there exist unique solutions for (O l )i and (0 2 ); i = 1, J<). (iv) Execute steps (d) to (g) of the trajectoryresolution procedure described in Section 3. * If all constraints are satisfied, it implies that the solution is a feasible one, but not necessarily the global optimal one. In order to obtain a global solution, the optimization repeats steps (i) to (iv) described above , by attempting to determine the optimal set of (a,b) coefficients . Thus , no matter how large K is , the optimization is always two- dimensional , searching for optimal (a, bt. (However, as it will be decri bed later and shown by an example this solution procedure is only an approximation of the global solution for the placement of [C]i , i = 1, J<) . * If anyone of the constraints (described in Section 3) is not satisfied, K is increased and the procedure described above, (i) to (iv), is repeated for the new K locations (i.e ., for a new set of iso-pos curves).

4. OPTIMAL PLACEMENT OF £(t)

In this section, an optimization procedure will be presented for the optimal placement of [C(t)]; , i = 1, J< , in the feasible search space. The proposed technique yields a approximation of the true globally optimal solution. The result is a sequence of locations with respect to Fw. First , we will consider the simplest of the cases and then generalize it to the most general two-arm system.

4.1. Optimization Procedure

Let us consider a two-arm planar system consisting of rotary joints only. Arm-1 carries the tool and has two dof. Arm- 2 carries the work piece and has one dof. If the orientational constraint is neglected at the contact point, then we note that the system has one degree of redundancy. The tool is required to track a trajectory on the object bounded by the points A and B, (i.e., the specific edge of the work piece shown in Fig. 2). When the tool is in contact with the object at any arbitrary point, [C(tO)] = (x(tO), y(tO)), the closedloop chain can be seen as a four-bar mechanism with a mobility of one dof. For this specific two-

The above procedure can be further improved by increasing the order of the curve selected to intersect the iso-pos curves (i.e., using an nth order polynomial , where n > 1) . The number of dimen927

sions of the optimization is however only n + 1 and remains independent of the number of locations chosen on the tool trajectory, K. For example, if one chooses a third-order polynomial for the intersecting curve, instead of a simple straight line, the optimization determines the global optimal set of four coefficients. This solution, however, is once again an approximate one, where the true global solution for the placement of [C]i can be achieved only when n = I< - 1. The flowchart shown in Fig. 3 summarizes the optimization technique presented above.

can be viewed when plotting the iso-Ioc curves, one has to note that every point on the curve is defined by a two-axis frame Ft expressed with respect to Fw by (x, y, a). It can be noticed that the search space in this case is a subset included in the search space obtained when orientational constraint is neglected , (namely limited segments of the iso-pos curves in Section 4-1) . Using the iso-Ioc curve definition above, the optimization procedure can be carried out for this case as well without any modification . Cubic-spline curves would be fitted to four joint variables, as a function of time, for K nodes. The treatment of constraints other than the orientation is the same as before.

~;~~itl:~ ~~~~~~:J~~~~~

K. '

K=K+ I

Position Constraint for Spatial Systems. The two-arm robotic system must have a total of four dof. These dof can be distributed as 2-2 and 31. The search space for the optimization would consist of K iso-pos curves, which are spatial in nature. In this case, one could use an "intersecting surface" , as opposed to an intersecting curve used in the case of planar systems, in order to intersect the spatial iso-pos curves. Accordingly, the number of variables, (i.e., the coefficients of the polynomial equation describing the spatial intersecting surface), defines the complexity of the optimization .

1----<.

Set up a search to detennine best coefficient values to minimize the motion-time

y"

Once the intersection points have been determined , the fitting of the cubic splines to the four joint variables , and thereafter determination of the optimal motion time for a feasible K- node approximation of the tool trajectory continues as described for the planar case.

Exit

Fig. 3. Optimization flowchart

In this paper, we suggest that a low-order polynomial intersection-curve (n ~ 3) would be quite sufficient , since the trajectory for Arm-l carrying the tool defined by the intersection points would be a smooth one and can be viewed as being very close to a 3rd order polynomial. Though it should be remembered that the actual trajectory is defined by the cubic-splines fitted to the joints variables .

Position and Orientation Constraints for Spatial Systems . The two-arm robotic system must have a total of seven dof. Moreover , the dof have to be properly distributed (numerically) and organized (geometrically) to give an overall desired redundancy of 1. These dof can be distributed as 4-3 , 5-2 and 6-1 . This case is a combination of cases (4.2 .1) and (4.2 .2) described above , specially in regards to the definition of iso-loc curves and the intersecting surfaces. (In the spatial case , the orientation of Ft with respect to Fw is decribed by three angles , as opposed to only one, a, needed in the case of planar systems).

4 .2. Generalisation for i-dof Redundancy Position and Orientation Constraints for Planar Systems. The two-arm robotic system must have a total offour dof. We assume that the robots cannot individually be redundant. These dof can be distributed as 2-2 or 3-1. The latter is achieved by adding 1 more dof to the 2-dof arm in the previous system , where the former is achieved by adding 1 extra dof to Arm- 2 which carries the object. In both cases , however , the search space for the optimization consists of K "iso- Ioc" curves corresponding to K locations, ( "loc" standing for location: position and orientation) on the tool trajectory. Although only two-dimensional curves

4.3. Generalisation for 2-dof Redundancy

In this case , the optimization procedure developed for I-dof redundancy case has to be modified. Also , the optimization may rely on input from the (human) planner in regards to the definition of the search space and starting point of the optimization.

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Position and Orientation Constraints for Planar Systems. The two-arm robotic system must have a total of five dof, which must be distributed as 3-2. The 4-1 dof distribution, though a valid one, will not be considered herein since the 4-dof arm is (individually) a redundant one. As can be visualized from Fig. 4, any contact point on the workpiece can be placed within an "iso-loc region" (as opposed to an iso-loc curve). For a K-node approximation, there would be K iso-pos reglOns, which might be partially overlapping.

J!!!-

Fig. 5. Locations of mid-points

,. ,"'. . . I ~-Ioc r~on

Position Constraint for Spatial Systems . In this paper, for spatial systems with more than I-dof redundancy, only the position constraint will be considered . As might have been already noted in the I-dof redundancy case, the consideration of orientation constraints only complicates the definition of the search space used during the optimization, but not the nature of the optimization. Thus, as in the case of planar systems, increasing the number of redundant dof will increase the dimensions of the iso-pos regions first to two and then to three .

(

l ~ lOC r'99l on e

Fig. 4. Iso-loc regions

The objective of the optimization is to determine the optimal set of K points . When we consider the earlier proposed technique of using a (polynomial) intersecting curve, we ncte that the intersections are specific individual segments of the polynomial curve , as opposed to being individual (intersection) points . A possible solution to this moderately complex optimization problem is to conduct a K -dimensional search for the location of the K nodes for every "guessed" polynomial. The equivalent problem is searching freely for the K-node locations directly within their 2-dimensional isopos regions. As can be noted, however, in both cases the optimization is directly coupled to the number of nodes , which is an undesirable scenario. The following approximation is proposed herein to significantly simplify the optimization procedure :

Let us consider the 3- 2 dof distribution case as an example. The K contact points on the tool trajectory could be placed anywhere within the K iso-pos regions, which are 2 dimensional and spatial in nature. Thus, by simply extending the earlier concepts, one can propose first the use of "polynomial surfaces" intersecting the iso-pos regions, and second the use of mid-point selection process, for finding K optimal points on the individual segments.

4.4 . Generalization for 3-dof Redundancy Position and Orientation Constraints for Planar Systems. In this case, both arms have full mobility (i .e., 3 dof each). The only difference with the 3- 2 dof distribution case is the definition of the iso-loc regions. Herein , these regions would be more expanded due to the extra dof.

(1) Determine the iso-loc regions for K nodes (or simply their boundaries) . (2) Guess a polynomial curve and let it pass through all the iso- loc regions . (3) Consider each region individually, by first determining the segment of the polynomial curve within it and then simply finding the mid-point of the segment , Fig. 5. (4) Solve inverse kinematics at the K mid- points. (5) Fit cubic- splines . Check for constraints and find the best objective function value attainable. Return to step (2) for guessing a different polynomial , and repeat optimization until "best" intersecting polynomial is determined .

Position Constraint for Spatial Systems . As in the case of the planar systems, both arms have full positional mobility (i.e. , 3 dof each) . The only difference herein is that the iso-pos regions now are 3-dimensional. In this case , the solution of the optimization problem would require the construction of a hierarchy. In the upper level, the optimization would guess a polynomial surface that would intersect the "iso-pos volumes" yielding a set of pseudo iso-pos surfaces . The best set of K node positions on these surfaces can be found as explained before within the second level of the hierarchy. The procedure would then return to the upper level to determine the best polynomial (intersecting) surface and its corresponding optimal set of K node locations.

The above procedure 's dimensionality is independent of K. It is only dependent on the order of the polynomial. The accuracy of the optimization can be further improved without a significant increase on the computational time by shifting the "constant" overall "string" along the polynomial curve, Fig. 5. 929

the motion. As shown in Fig. 7, f{ = 5 nodes were necessary to satisfy the position and speed errors. Once K was set at 5, the motion-time was minimized, by reducing the constant-time intervals between the nodes, to as low as 0.59 secondes. The "flexible polyhedron" search method of Nelder and Mead (Nelder and Mead, 1964) for non-linear optimization problems, was succesfully used for determining the two optimal coefficients of the first-order polynomial used to intersect the iso-pos curves. When the order of the polynomial was increased to first n = 2 and then n = 3 the (reduced) optimal motion times were determined as 0.50 sec and 0.44 sec respectively. Figures 8 and 9 show that both position and speed errors remain within their specified tolerances for K 2: 5. The optimization results , in regards to positional and speed errors, are summarized in Table 2. The motion simulation for the 3rd order intersection polynomial is shown in Fig. 6.

4.5. Discussion

The complexity of our proposed method increases significantly as the "dof of redundancy" increases. However, in the case of modular robots , there is no need for "high" number of redundant dof, where one redundant dof is sufficient . Thus, for high numbers of redundant dof, one may choose to use an "exact" technique as in (Tabarah et al. , 1992) (one trajectory is arbitrarily chosen for the arm carrying the tool and the conjugate trajectory is solved for the second arm) , and not an approximation technique as discussed in this paper.

5. EXAMPLE In order to illustrate the proposed algorithm , the two-arm planar robot shown in Fig. 2 was considered . The tool trajectory was approximated by K points. Positional errors were calculated by determining indi vid ual points on the (simulated) actual tool trajectory and finding their shortest distances to the designated tool trajectory. Similarly, the velocity vectors were determined at those "paired" points , and errors in relative speed were calculated as a percentage of the desired pre-set constant speed that the tool had to follow along the desired tool trajectory on the object.

Table 1 Robots' parameters

Arm Joint

qmax(rad/s) qmax(rad/s 2 ) Link Length (m)

Motion- time was considered as the objective function for the optimization. This motion time can be calculated from the robot 's kinematics as follows:

System's

IBij(s)1 ()r;ax

1 2 l.0 20 .0 0.8

2 1 0.5 10.0 0.5

toAotTon

Robo! I

max max -.- - , max

[ ij ,.

1 1 0.5 10.0 0.8

ij ,3

O:Ss:S 1;i= 1, 2;j= 1, 2

(1)

where s is a time scale, Bij(S) and Bij(S) are the first and second derivatives of the joint displacement obtained from the cubic- spline interpolaax and B rn ax are the velocity and actl'on , and Brn 'J 'J celeration limits on the ith joint of the jth arm.

0 . 0

X

The overall optimization , which contained total motion simulation, was first run using a firstorder polynomial for the intersection curve. The data used during the simulations are given in Table 1 where the base locations of the two arms were 's et as .c.~ = (0 , 0) and .c.~ = (l.7m, 0) , Fig. 6. The object held by Arm-2 was rectangular in shape (0.5rn x 0.2m), whose longer edge constituted the desired tool trajectory. The positionerror limits were set as ± 0.5 mm and the speederror limits were set as ± 1 % of the desired "constant" tool speed. Since the motion- time was minimized , the desired relative tool velocity was also minimized, but maintained constant during

(m)

Fig. 6. System 's layout and optimal motion: K yd order intersecting polynomial

= 5,

Table 2 Maximum position and speed errors and motion times for first , second and third order polynomials as a function of K

Number 01 Nodu (K )

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\01

3'"

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0.'3

O.9!I

DlS

ua

0.17

1 60

O~7

0 43

Tunt

it )

~':...I ~( .., TIIT'Ie It )

0 41

O.6t

0..59

030

O.~2

060

0. 44

091

0-X1

OJl

079

0..51

0 "9

100

0.4-4

0 42

0.99

0 45

It should be remembered that the time intervals between the nodes were reduced during the optimization while considering the limits on joint velocities and accelerations.

was discussed for higher numbers of dof redun-

dancy. An optimization procedure minimizing a given cost function (the robot motion time in our case) for this pair of trajectories was described. For the overall synthesis problem, it will be necessary to consider other variables during the optimization, for example, relative distance between the two bases or the link lengths. The use of the dynamics model of the system is also necessary to take into account the dynamics constraints .

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7. REFERENCES Fig. 7. Position and speed errors - first order polynomial

Cohen, R ., M.G. Lipton , M.Q. Dai and B. Benhabib (1992). Conceptual design of a modular robot. ASME, Journal of Mechanical Design 114, 117-125. Craigan, C.R. and D.L. Akin (1989). Optimal force distribution for payload positioning using a planar dual-arm robot. ASME, Journal of Dynamical System, Measurement and Control 111 , 205- 210 .

••

o ••

0

.0

, . .... c. )

••• •

.0

o •

•

0

Tl .. ,

.0

c. )

••

Jouaneh, M.K., Z. Wang and D.A. Dornfeld (1990). Trajectory planning for coordinated motion of a robot and a positioning table: part 1 - path specification. IEEE Journal on Robotics and Automation 6 , 735- 745. Nelder , J. A. and R . Mead (1964). A simple method for function minimization. The Computer Journal 7 , 308-313. Tabarah , E. , B. Benhabib, R.G. Fenton and G. Hexner (1992). Optimal trajectory resolution for the coordination of two robots in contact operations. Proc . ASME 22nd Biennial Mechanisms Conference 45 , 455-46 0. Tabarah , E., V. Taussac , E. Dombre and B. Benhabib (1993). Optimal trajectory planning for coordinated two-arm modular robotic systems. Proc. International Conference on Advanced Robotics pp. 617-622. Tarn, T.J., A.K. Bejczy and Z.F. Li (1988). Dynamic workspace analysis of two cooperating robot arms. Proc. A merican Control Conference 1 , 489- 498. Tesar , D. and M.S. Butler (1989). A generalized modular architecture for robot structures . ASME, Journal of Manufacturing Review 2 , 91-117. Zheng, Y.F. and J.Y.S. Luh (1988). Optimal load distribution for two industrial robots handling a single object. Proc. IEEE International Conference on Robotics and Automation pp . 344349.

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Fig. 8. Position and speed errors - second order polynomial

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to ...

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Fig. 9. Position and speed errors - third order polynomial

It is clear from the simulations that higher-order

polynomial intersecting curves yield a set of K locations which better approximate the nominal task-space tool trajectory in terms of minimum task-execution times. The only disadvantage to higher-order functions is obviously the larger number of coefficients that have to be optimized.

6. SUMMARY

The concept of using modular systems to configure two-arm robots was introduced. The two arms are coordinated to move relative to each other in order to perform a common task in an optimal manner. A novel method was presented for the planning of the coordinated motion for a 1-dof redundancy case. This method allows to determine a pair of conjugate trajectories for the two end effectors. An extension of this method 931