Advances in Robot Kinematics - Jadran Lenarcic and Bernard Roth (Eds) Part 11 potx

QUANTITATIVE DEXTEROUS WORKSPACE COMPARISONS Juan A. Carretero and Geoff T. Pond Department of Mechanical Engineerin g, University of New Brunswick 15 Dineen Dr., Fredericton, NB, E3B 5A3, Canada {Juan.Carretero, Geoff.Pond}@unb.ca Abstract singanovel method for the formulation of Jacobian matrices, this paper will compare the dexterity of two parallel manipulators: the 3- P RS and 3-RPS. The newly obtained Jacobian matrix is square and dimensionless and its singular values haveanevident physical meaning. These singular values are used to identify and compare regions of the manipulators’ workspaces where either high end e ector velocities or a fine resolution over the manipulator pose may be obtained.Workspace plots for the two manipulators, corresponding to these attributes are presented and compared for arbitrarily chosen architectural parameters. The objective of the comparison is to illustrate the capability of the described method to quantitatively compare the dexterity of various complex degree of freedom manipulators. Keywords: Parallel manipulators, Jacobian matrices, singular value decomposition, 1. Introduction Studying the workspace characteristics of manipulators having either translational degrees of freedom (DOF) or rotational DOF is simplified due to the dimensional consistency within the manipulator’s Jacobian matrix (e.g., Tsai and Joshi, 2000). However, for manipulators having DOF in both translational and rotational directions, the conventional method of studying the workspace characteristics using the Jacobian matrix condition number is no longer possible. This is due to dimen- sional inconsistencies within the manipulator’s Jacobian matrix. Recently, a method has been introduced by Pond and Carretero, 2006, which produces a dimensionally homogeneous Jacobian matrix regard- less of the manipulator’s degrees of freedom, provided that only one type of actuator (either revolute or prismatic) is used in the manipu- lator architecture. This paper will further this work by quantitatively comparing the dexterity of different manipulators. The manipulators included in this study are the 3-P RS mechanism (Carretero et al., 2000) and the 3-RP S mechanism (Lee and Shah, 1988). © 2006 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 297–306. 297 workspace, dexterity, complex degrees of freedom U The two mechanisms may be considered to have the same independent DOF. The resulting Jacobian matrix for both manipulators, when for- mulated by the method used in this paper, are the same size and are dimensionless. This will be explained further in the following section. 1.1 Jacobian Formulation A detailed description of the method developed to obtain dimen- sionally homogeneous Jacobian matrices for use in dexterity analyses is described in Pond and Carretero, 2006 and therefore, only a brief introduction is provided here for completeness. The method presented therein, is a continuation of a method introduced by Gosselin, 1992, and later generalised by Kim and Ryu, 2003. This original method relates a vector containing the Cartesian velocities of three points on the end effector platform ( ˙ G 1 , ˙ G 2 , and ˙ G 3 ) to the vector ˙q of actuator velocities using ˙q = J˙x where ˙x =[ ˙ G 1 x ˙ G 1 y ˙ G 1 z ˙ G 2 x ˙ G 2 y ˙ G 2 z ˙ G 3 x ˙ G 3 y ˙ G 3 z ] T , ˙q =[ ˙ b 1 ˙ b 2 ˙ b 3 ] T and the Jacobian matrix J relating the two is: J =  dq dx  (1) This method relates actuator and end effector velocities for up to 6-DOF motion. However, use of this Jacobian’s condition number or singular values in dexterity analyses may not be appropriate as at least 3 of the variables of ˙x are dependent on the remaining terms of ˙ x (6 are dependent in the case of 3-DOF manipulators). Therefore the meaning of the Jacobian’s singular values is unknown (Kim and Ryu, 2003). Now consider the case of a 3-DOF manipulator such as one of those included in this study, having a translational degree of freedom per- pendicular to the base platform and two rotational degrees of freedom around axes parallel to the base platform. Knowing the elevation of three points on the end effector platform relative to the base platform is suf- ficient to solve the inverse displacement problem (Pond and Carretero, 2006). Therefore, vector ˙x may be reduced to: ˙x  =[ ˙ G 1 z ˙ G 2 z ˙ G 3 z ] T and the general equation relating these end effector velocities to the actuator velocities is rewritten in the form ˙q = JP ˙x  where P, which is used to map ˙x  to ˙x, is expressed as: P =  dx dx   (2) In order to obtain the partial derivative elements of P, constraint equa- tions relating G i z to G i x and G i y must first be formulated. The first derivative with respect to time of these equations leads to the partial 298 J .A. C arretero an d G .T. Pon d derivative elements of P. In the case of the manipulators included in this comparison, where the actuated joints are all prismatic, the multiplica- tion of JP yields a square, dimensionless Jacobian relating independent end effector velocities to the actuator velocities. That is, JP =  dq dx   (3) Therefore, the singular values of such a Jacobian matrix as JP in Eq. 3 have an evident physical meaning. The meaning of which is important in understanding the dexterous characteristics of the manipulator. 1.2 Dexterity Analysis The dimensionally homogeneous matrix JP maps the system output ˙x  to the system input ˙q. As such, useful system characteristics may be extracted from the information contained within this matrix. As explained by Pond and Carretero, 2006, the maximum and minimum singular values of JP correspond to the magnitude of the minimum and maximum system outputs corresponding to any unit system input. Therefore, for the case where relatively small singular values are ob- tained from JP, this corresponds to a manipulator pose where high end effector velocities may be obtained for a given set of actuator veloci- ties. Conversely, where relatively large singular values are obtained, a finer resolution over the manipulator pose results as the system output is smaller for the same unit input. Conventionally, the Jacobian matrix condition number is used to mea- sure the dexterity of a particular manipulator pose. Mathematically, this is the ratio of maximum and minimum singular values. Physically, this compares the ease by which the manipulator may move in the fastest and slowest directions or whether the manipulator has similar resolution in each of the DOF. Poses where the condition number is exactly 1 are termed isotropic configurations (Angeles, 2003) and are considered ideal. Only by examination of both, the Jacobian’s condition number and its singular values, can a true understanding of the workspace characteris- tics of a given manipulator be achieved. 2. The 3-PRS Mechanism The 3-PRS, shown schematically in Fig. 1a consists of three identical kinematic chains. The kinematics for this manipulator have been previ- ously presented in a variety of publications (e.g., Carretero et al., 2000) and therefore only a brief review is provided here. Q uantitative Dexterous Wor k space Comparison s 299 Figure 1. Basic structure of the 3-P RS (a) and 3-RPS (b) manipulators Inverse Displacement Solution. Conventionally, the two mecha- nisms discussed in this paper have one translational DOF along the base frame’s z-axis and two rotational DOF around the x and y axes depicted in Fig. 1a. However, as mentioned earlier, these degrees of freedom may also be modelled using the elevation of three points lying on the plane representing the moving platform (G 1 z , G 2 z ,andG 3 z ). Each of these three points is chosen as one of the two intersections of the circle concen- tric and coplanar with the end effector platform, with the plane defined perpendicular to the fixed xy plane. In order to compare between differ- ent architectures, the magnitude of the circle’s radius where points G i lie must be kept constant, and is preferably a unit circle. Constraint equa- tions were developed in Pond and Carretero, 2006, relating G i z to G i x and G i y is very involved, they will not be presented here. As presented in Carretero et al., 2000, the displacement of the actu- ated prismatic joint |b i | is solved by means of the vector loop represent- ing limb i, depicted in Fig. 1a for i = 3. Given a platform elevation z and two angles ψ and θ around the fixed x and y axes, respectively, two solutions for this displacement are obtained by solving for the squared length of limb i as |l i | 2 = |r i − b i | 2 . Jacobian Formulation. As described by Pond and Carretero, 2006, any point on the plane defined by the moving platform, with respect to the base frame may be equated to a weighted sum of the three vectors g i (i = 1, 2, 3), depicted in Fig. 1a. A point must now be identified which lies on this plane and may also be represented by a vector loop which includes the actuated prismatic joint. The logical choice is the 300 J.A. Carretero and G.T. Pond by limb i,fori = 1, 2, 3. The plane defined by limb i is constrained to be (for i = 1, 2, 3). As the derivation of these constraint equations position of the spherical joint as this point obviously lies on the end effector plane and a vector loop including the actuated prismatic joint has already been used when solving for the actuator displacement. The vector r i , representing the position of the spherical joint with respect to the base frame, is equivalent to the vector sum b i + l i . Equating this to the weighted summation of vectors g i yields: b i + l i = k i,1 g 1 + k i,2 g 2 + k i,3 g 3 (4) where the variables k i,j (j=1, 2, 3) are dimensionless constants and k i,1 + k i,2 + k i,3 = 1. Finally, taking the first time derivative and simplifying: s T b i s l i ˙ b i = k i,1 s T l i ˙g 1 + k i,2 s T l i ˙g 2 + k i,3 s T l i ˙g 3 (5) where s b i and s l i are unit vectors in the directions of b i and l i , respec- tively. The dimensionless constants k i,j are solved using Eq. 4. Equa- tion 5 may be written three times corresponding to each of the three limbs to formulate the manipulator Jacobians: J q ˙q = J x ˙x (6) where ˙x 9×1 =  ˙g T 1 ˙g T 2 ˙g T 3  T , ˙q 3×1 =  ˙ b 1 ˙ b 2 ˙ b 3  T and J q = ⎡ ⎢ ⎣ s T b 1 s l 1 00 0 s T b 2 s l 2 0 00s T b 3 s l 3 ⎤ ⎥ ⎦ 3×3 J x = ⎡ ⎢ ⎣ k 1,1 s T l 1 00 0 k 2,2 s T l 2 0 00k 3,3 s T l 3 ⎤ ⎥ ⎦ 3×9 The Jacobian matrix, J in Eq. 1 is produced by multiplying J x and J q , i.e., J = J −1 q J x . By taking the first time derivative of equations constituting the inverse displacement solution discussed in Section 2, the constraining matrix P in Eq. 2 is formulated. Finally, multiplication of the Jacobian J with the constraining matrix P produces the constrained, dimensionally homogeneous, square Jacobian JP in Eq. 3. 3. The 3-RPS Mechanism The 3-RPS mechanism depicted in Fig. 1b has been previously studied in a variety of publications (e.g., Lee and Shah, 1988). Inverse Displacement Solution. Similarly to the 3-PRS Mecha- nism, the limbs of the 3-RP S Mechanism are also confined to move on a single plane. As such, the constraint equations relating the designated independent degrees of freedom (G 1 z , G 2 z , and G 3 z ) to the dependent degrees of freedom (G i x and G i y for i =1, 2, 3) are the same. The only Q uantitative Dexterous Wor k space Comparison s 301 difference from the inverse displacement solution of the 3-PRS mech- anism is in the final step where l i = r i − b i is used to solve for the displacement of the actuated prismatic joint i, that is |l i |. Jacobian Formulation. Due to similarity in the constraint equa- tions for the 3-PRS and 3-RPS mechanisms, the development of the dimensionally homogeneous Jacobian J is also very similar. Consider Eq. 4 again. The vector b i is a known constant in the architecture of the 3-RP S mechanism whereas the magnitude of vector l i is now a variable. Taking the first time derivative of Eq. 4 yields: ˙ l i = k i,1 s T l i ˙g 1 + k i,2 s T l i ˙g 2 + k i,3 s T l i ˙g 3 (7) Writing Eq. 7 three times (once for each limb) to formulate the inverse and direct Jacobian matrices, as in Eq. 6, yields J q = I 3×3 and J x = ⎡ ⎢ ⎣ k 1,1 s T l 1 00 0 k 2,2 s T l 2 0 00k 3,3 s T l 3 ⎤ ⎥ ⎦ 3×9 (8) The Jacobian matrix J is obtained by the multiplication J −1 q J x .As previously mentioned, the constraint equations relating G 1 z , G 2 z , and G 3 z to G i x and G i y (i =1, 2, 3) are the same as those developed for the 3-P RS manipulator and therefore matrix P is identical. The resulting matrix JP is, again, square and dimensionally homogeneous. 4. Dexterous Workspace Comparison The reachable workspace is defined here as all poses attainable by a manipulator without forcing it to transit a singular configuration when travelling from its datum position (explained later in this section). Be- fore determining the reachable or dexterous workspaces, an understand- ing of the manipulator singular configurations must first be obtained. Inverse singular configurations typically correspond to the absolute boundary of the reachable workspace, beyond which, poses are unattain- able by the manipulator. Therefore, these singular configurations do not further limit the reachable workspace volume as defined in this paper. Direct singular configurations on the other hand, may exist inside the reachable workspace. In this work, only poses attainable without forc- ing the manipulator to transit a direct singular configuration, or toggle point, are included as part of the reachable workspace. The 3-RP S and 3-PRS mechanisms share the same direct singular con- figuration where the vector l i is parallel to the plane defined by points A i . 30 2 J .A. C arretero and G .T. Pond Figure 2. Reachable and Dexterous workspace for the 3-PRS (a and b) and the 3-RP S (c and d). Note θ and ψ are in radians, r p =0.6, for the 3-PRS: γ =0 and |l i | = 1 and for the 3-RPS: |b i | =1. Dexterous workspace boundary at cond max =6. At this pose, the manipulator gains the ability to toggle about the spher- ical joint A i .Forthe3-PRS and 3-RPS manipulators, when all three of the limbs are parallel with this plane, the end effector is at the minimum permissible elevation along the z-axis here referred to as the datum po- sition. The 3-P RS has a second family of singular configurations which occur when one or more of the fixed-length links become perpendicular to the direction of their respective actuated prismatic joint. 4.1 As the condition number of the Jacobian matrix is infinity at a sin- gular pose, it is expected that dexterity decreases as the manipulator approaches a singular configuration. Both the reachable and dexterous workspaces for the 3-P RS mechanism are depicted in Figs. 2 a and b. The bottom apex of Fig. 2a corresponds to the direct singular configu- ration discussed earlier at z = 0. The dexterous workspace depicted in Fig. 2b is a subset of the reachable workspace. As the maximum per- missible limit on the Jacobian condition number is reduced (not shown), the cross sectional area of the dexterous workspace continues to be re- duced. The workspace will however, continue to exist throughout the entire range 0 ≤ z ≤ 1. This tendency confirms that the manipula- tor is at isotropic conditions whenever the moving platform is perfectly parallel to the base platform where the JP is a scaled identity matrix. Theoretically, the reachable workspace of the 3-RP S mechanism ex- tends to a z-elevation of infinity. However, in order to provide a mean- ingful comparison to the 3-P RS mechanism, the range of z-elevations will be restricted to 0 ≤ z ≤ 1. This limited workspace is depicted in Fig. 2. As the 3-RP S mechanism also experiences the toggle point at z =0, the restricted range does not further limit the workspace on the lower Quantitative Dexterous Workspace Comparisons 303 Dexterity Measured by the Jacobian Condition umber N o −1 0 1 −1 0 1 0 0.2 0.4 0.6 0.8 1 ψ PRS: σ ≥ 0.25 (V = 0.0632) θ z −1 0 1 −1 0 1 0 0.2 0.4 0.6 0.8 1 ψ PRS: σ ≤ 0.85 (V = 0.0366) θ z a) b ) −1 0 1 −1 0 1 0 0.2 0.4 0.6 0.8 1 ψ RPS: σ ≥ 0.25 (V = 0.3439) θ z −1 0 1 −1 0 1 0 0.2 0.4 0.6 0.8 1 ψ RPS: σ ≤ 0.85 (V = 0.1503) θ z c) d ) Figure 3. Regions of the workspace corresponding to high end effector velocities and fine resolution over the manipulator pose for the 3-P RS (a and b, respectively) and the 3-RP S (c and d, respectively). Angles θ and ψ are in radians. boundary. Only the highest z-elevation is being imposed, which artifi- cially limits the range of both the reachable and dexterous workspaces. Otherwise, the cross sectional area of the workspace would monotoni- cally expand until, at z = ∞, the manipulator would be capable of 90 ◦ rotations around both the x and y axes (angles ψ and θ, respectively). It should be noted that neither of these two architectures has been optimised. That is, the values chosen for architectural parameters such as the base and end effector platform radii, and the magnitude of the fixed length link have not been optimised to provide the largest possible workspace volume. The objective here is only to illustrate the ability of this method to compare the dexterity of various architectures, not to suggest any single architecture is in any way superior to others. 4.2 Dexterity easured by both the Jacobian ondition umber and ingular alues One of the greatest advantages of the comparison method presented in this paper is the ability to identify regions of the manipulator’s workspace where either high end effector velocities may be achieved, or where a fine resolution over the manipulator pose exists. Depending on the applica- tion, either or both of these attributes may be highly desired. In the previous section, the condition that cond(J) ≤ 6 was arbitrarily chosen to determine the dexterous workspace. In this section, the dexterous workspace plots are further reduced by constraining the singular values of the Jacobian matrix to be within a defined range. In this way, regions of the workspace corresponding to either high end effector velocities, or fine resolution over the manipulator pose, are determined. For the dexterous workspace of the 3-P RS manipulator, depicted in Fig. 2b, Jacobian matrix singular values vary within the range 0.003 ≤ 304 J .A. C arretero and G .T. Pon d M C N S V Figure 4. Workspace volume as a function of a minimum (left) and maximum (right) permissible singular values. σ ≤ 15.565. Figure 3a is produced by restricting the Jacobian condition number to be less than or equal to 6, and restricting all Jacobian matrix singular values to be greater than or equal to 0.25. That is, for a system output where | ˙x| = 1, only poses where the magnitude of the vector ˙q is greater than 0.25, are allowed. Therefore, the workspace is restricted to poses where a minimum resolution over the end effector pose is allowed. Similarly, Fig. 3b is produced by restricting the Jacobian matrix con- dition number to be less than or equal to 6, but in this case, only sin- gular values less than or equal to 0.85 are allowed. In this manner, the workspace is restricted to poses where the of the vector ˙q is restricted to be less than 0.85, for any unit system output or, | ˙x| =1. Intuitively, this is not in the same region of the reachable workspace as poses were a fine resolution is obtained. Figures 3a and 3b confirm this as these two regions are at opposite ends of the workspace. Similar limits on the 3-RP S manipulator were used to obtain Figs. 3c and 3d. Figure 3c shows that the 3-RP S has lost a relatively small volume by restricting the Jacobian matrix singular values to be less than or equal to 0.25. This should not be surprising as the permissible range 0 ≤ σ ≤ 0.85 is a large portion of the overall range 0.0007 ≤ σ ≤ 1.0526 observed when obtaining Fig. 2d. The left portion of Fig. 4 compares the two manipulators’ workspace volume as a function of the minimum permissible singular value. As this value is increased, an increasingly heavier emphasis is placed on high end effector velocities. Naturally then, as the σ min is increased, a greater restriction is placed on the workspace and the workspace volume continues to decrease in magnitude. The 3-P RS mechanism experiences a much more gradual reduction in workspace volume as the minimum permissible singular value is decreased. This demonstrates its ability to achieve a high degree of accuracy throughout a relatively large portion of its workspace as compared to the 3-RP S. Similarly, a gradual increase in the workspace volume is experienced by the 3-P RS mechanism as the maximum permissible singular value is increased, when compared to the 3-RP S (see the right portion of Quantitative Dexterous Workspace Comparisons 305 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 σ σ min Volume 3 PRS, γ = 0 3 RPS 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1  max Volume 3 PRS, γ = 0 3 RPS Fig. 4). It is observed that, as σ max is decreased, the workspace size is further restricted by a continuously heavier emphasis on the accuracy of the manipulator. It should be noted that the significantly smaller dexterous workspace volume of the 3-P RS manipulator is due in part, to the considerably wider range of singular values within the manipulator’s reachable workspace when compared to the 3-RP S manipulator. 5. Conclusions The method for developing dimensionally homogeneous square Jaco- bians JP has proven successful in comparing the dexterity of multiple parallel architectures. This method is suitable for comparing manipula- tors having the same number and type of actuators and also having the same independent degrees of freedom. For illustration purposes, the 3-P RS and 3-RPS manipulators were compared using an arbitrary set of architectural parameters for each manipulator. Of the two, the 3-P RS manipulator was found to have the largest workspace if a high level of accuracy is required and the 3-RP S manipulator was found to have the largest workspace if high end effector velocities are required. Following optimisation, any of the manipulators may be the best candidate depending on the designer’s desired compromise between the two characteristics. Having identified a desired compromise between accuracy and veloc- ity characteristics, the robot designer may use this method to optimise and compare a variety of manipulators in order to select the optimal architecture and architectural variables, for a specific application. References Angeles, J. (2003), Fundamentals of Robotic Mechanical Systems, Springer- erlag. Kinematic analysis and optimization of a new three degree-of-freedom spatial par- Gosselin, C.M. (1992), The optimum design of robotic manipulators using dexterity Kim, S. G., and Ryu, J. (2003), New dimensionally homogeneous Jacobian matrix formulation by three end-e ector points for optimal design of parallel manipulators, Lee, K. and Shah, D. (1988), Kinematic analysis of a three-degrees-of-freedom in- Pond, G., and Carretero, J.A. (2006), Formulating Jacobian matrices for the dexterity analysis of parallel manipulators, To appear in Mechanism and Machine Theory. Tsai, L.W.,andJoshi, S. (2000), Kinematics and optimization of a spatial 3-P 306 Carretero, J. A., Podhorodeski, R.P., Nahon, M.A., and Gosselin, C.M. (2000), J.A. Carretero and G.T. 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(1999), Determination of 6D workspaces of Gough-type parallel manipulator and comparison between different geometries, The International Journal of Robotics Research, vol 18, no 9, pp 902–916 Merlet, J-P (2004b), ALIAS, [Online] Available: http://www-sop.inria.fr/coprin/ logiciels/ALIAS/ Ming, A., and Higuchi, T (1994), Study on multiple degree-of-freedom positioning mechanism using wires (Part 1) - concept,... and reconfigure and they have the possibility of working in a very large space Consequently, parallel cable-driven mechanisms have been used in several applications such as, for instance, robotic cranes (Dagalakis et al., 1989), high speed manip- 315 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 315–322 © 2006 Springer Printed in the Netherlands 316 M Gouttefarde et al A8 8 6 4 A7 A5 A4 y... and the eight solutions of complex legs A, B and C we obtain 133 = 2197 distinct solutions of fully-isotropic PMs with six degrees of freedom All these solutions obey conditions (2 )-( 10) and integrate only revolute and prismatic joints in the legs A, B and C The family of these solutions could be more diversified by introducing helical, cylindrical and planar joints to replace some equivalent combinations... matrix of fully-isotropic hexapods presented in this paper is the identity matrix throughout the 323 J Lenar i and B Roth (eds.), Advances in Robot Kinematics, 323–330 © 2006 Springer Printed in the Netherlands 324 G Gogu entire workspace (J = I6× 6) The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission... three independent translations of the intermediary platform n1 nA nB nC and implicitly of the final mobile platform 5F They make a fully-isotropic translational parallel kinematic chain The legs D, E and F contribute to the three independent rotations of the mobile platform 5F and each integrates two homokinetic joints connected by telescopic shafts Just the input and the output shafts are indicated in. .. determine the points of intersection between Ci and the other cubic curves The points of intersection between the cubic curves C i can be determined by means of an elimination method (Roth, 1993) Step 3: for each of the cubic curve Ci , find all its parts defined by the points of intersection computed at step 2 and eliminate all unbounded parts 320 M Gouttefarde et al Determination of the Wrench-Closure . mechanism (Carretero et al., 2000) and the 3-RP S mechanism (Lee and Shah, 1988). © 2006 Springer. Printed in the Netherlands. J. Lenarþiþ and B. Roth (eds.), Advances in Robot Kinematics, 297–306. 297 workspace,. Pond Figure 2. Reachable and Dexterous workspace for the 3-PRS (a and b) and the 3-RP S (c and d). Note θ and ψ are in radians, r p =0.6, for the 3-PRS: γ =0 and |l i | = 1 and for the 3-RPS: |b i | =1 the cross-section boundary curve. Regions and Sub-Regions, on Advances in Robot Kinematics, Ljubljana, pp. 15 0- International Workshop on Compu- World Congress on the Symp. on Theory and Practice