Re-design of Force Redundant Parallel Mechanisms by Introducing Kinematical Redundancy

Transcription

1 The 2009 IEEE/RSJ International onerence on Intelligent Robots and Systems October 11-15, 2009 St. Louis, USA Re-design o Force Redundant Parallel Mechanisms by Introducing Kinematical Redundancy Kiyoshi Nagai and Zhengyong Liu Department o Robotics, Ritsumeikan University, Noji-higashi 1-1-1, Kusatsu, Shiga, JAPAN nagai@se.ritsumei.ac.jp Abstract Force redundant mechanism is adopted to achieve high acceleration. However, utilizing the good velocity control perormance o servo motor to achieve a high positional accuracy will arise ecessive big internal orces. Thereore, an idea o introducing kinematical redundancy into orce redundant parallel mechanism is proposed. In this idea, we do not add kinematical redundancy on the sub arm but introduce internal redundant motion into the top plate. The concept o this research is illustrated by improving a simple orce redundant parallel eample, and the merits o the improved mechanism are summarized. This theory is applied into modiying the real high speed parallel mechanism NINJA. Its kinematics and static orce relationship are derived and compared respectively. The prototype o the new mechanism is proposed. I. INTRODUTION We aimed at designing a ast robot that achieves an acceleration perormance over 100 G] together with high precision. High speed and high precision positioning mechanisms are required in many applications, or eample, or bonding semiconductor-tip terminals with a lead line. This procedure demands high speed and accuracy in a narrow workspace. Researches related to high speed motion and high precision can be ound in 1]2]3]4]. Parallel mechanism have many ecellent characteristics in terms o speed, accuracy, stiness, and load carrying capacity compared to their serial counterparts 5]6]. Some o high speed mechanisms using the advantages o parallel manipulators, such as DELTA7], HEXA8], FALON9] and RPlAH 10] have been developed, but these mechanisms did not achieve perormances over 100 G]. On the other hand, singularities are common in parallel mechanisms. They will bring disadvantages to parallel mechanisms 11]12]. Then orce redundancy is introduced into parallel mechanism as it is an eicient way to reduce or even eliminate these singularities rom parallel manipulator 12]13]14]. Besides, orce redundant mechanism also has the nice eatures o increasing the reliability o the system, improving artesian stiness, achieving more homogeneous output orces, increasing acceleration and minimizing internal loading torques o the actuators o the manipulator 15]16]17]. As a result, some high speed parallel mechanisms with orce redundancy appeared. For eample, Kock and Schumacher presented a planar parallel manipulator PA- R-MA, which is designed or high-speed pick-and-place application. And eperimental results show that, or position tracking, tracking errors were less than 0.5 mm in X and Y directions with acceleration o 70 m/s 2 and velocity o 3 m/s 18]. By utilizing the good perormance o parallel orce redundant mechanism, we presented a ast parallel mechanism NINJA, which can reach the highest acceleration up to 100 G], in 19]. However, orce redundancy mechanism will make velocity o active joints not independent. As a result, it is hard to get a high positional accuracy by adopting velocity control scheme or servo motor because it has the risk o producing big ecessive internal orces 16]. Thereore, we are introducing kinematical redundancy to redesign a orce redundant parallel mechanism NINJA. In this designing, we do not add kinematical redundancy on any sub arm but between the sub arms and top plate. Hence, all the sub arms do not need any change but a new top plate is proposed. Due to introducing kinematical redundancy into the orce redundant parallel mechanism, joint variable is no longer dependent and the accuracy perormance in position control o the modiied one is epected to be improved while keeping the high acceleration simultaneously. This paper is organized as ollows. In Section 2, a simple orce redundant parallel mechanism is used to state the problem. The concept o the research is illustrated by inding the solution to improve it. In Section 3, basic equations o modiying general orce redundant parallel mechanism by introducing internal kinematical redundancy are developed and the design conditions are given. In Section 4, this proposed theory is implemented to redesign the real mechanism NINJA. The new structure and kinematic equations are given. In Section 5, prototype o the new mechanism NINJA has been developed. Finally, conclusion and uture work are given in Section 6. II. ONEPT ON INTRODUING KINEMATIAL REDUNDANY A simple orce redundant parallel manipulator is analyzed and the problem o orce redundant mechanism is described. Then by inding the solution to this problem, the concept o adding internal kinematical redundancy between sub arms and the top plate is described. A. Problem Statement The simple orce redundant parallel manipulator without internal kinematical redundancy is shown in Fig.1 It is a 1 DOF parallel manipulator with 1-dimensional workspace (linear motion) and 2 actuators in parallel. The nomenclatures are shown as ollows: /09/$ IEEE 5898

2 Sub arm 1 S 1 Sub arm 1 S 1 q 1 Top plate q 1 θ q A2 2 S2 Sub arm 2 s 1 o s2 Fig DOF orce redundant parallel manipulator without internal kinematic redundancy q = q 1 q 2 ] T: Joint variables. qi is the active joint o the i-th sub arm (i = 1,2) S i : The connection point between the i-th sub arm and the top plate s = s 1 s 2 ] T, si is Position o S i along ais with respect to O : Position o top plate along ais with respect to O A = A2 ] T : The set o actuator driving orces. : Generalized orce at the top plate As shown in Fig.1, there are two ways to calculate the connection point velocity. The irst route is to set up the velocity relationship between connection point and joint variables, i.e. ṡ 1 = q 1, ṡ 2 = q 2 (1) Using vector notation to write (1) is 10 ṡ = J S q, J S = (2) 01 J S is the manipulator Jacobian matri transorming rom joint velocity to the velocity at the connection point. The other route is the velocity relationship between connection point and the top plate, that is Equation (3) is written compactly as ṡ 1 = ẋ, ṡ 2 = ẋ (3) ṡ = J T ẋ, J T = 11 ] T J T is the manipulator Jacobian matri which relates top plate velocity to the velocity at the connection point. ombining (2) and (4) reduces the kinematics as ollow: q = Jẋ, J = J 1 S J T = 11 ] T (5) J is the Jacobian matri mapping top plate velocity to joint velocity. Thus statics o this simple mechanism is: = J T A = 11 ] = + A2 (6) A2 (4) q 2 o A2 R Sub arm 2 s 1 s 2 S 2 New top plate Fig DOF orce redundant parallel manipulator with internal kinematic redundancy In this orce redundant mechanism, and A2 can be dierent or a given which is the sum o and A2, as shown in (6). However, q 1 and q 2 can not have dierent values as it is decided by ẋ, as shown in (5). As a result, the velocity control unction o the actuators can not be applied. In other words, because o the particularity o orce redundant mechanism, it is required that a set o joint variables must be dependent otherwise big internal orces will arise while pursuing high accuracy by using velocity control scheme or each actuator. Taking this mechanism shown in Fig.1 as an eample, the velocity o each sub arm must be the same at any moment. I there is any small dierence between them, the internal orce between sub arm and the top plate will be very large and the mechanism may break down. This is a big problem. The requirement o no position dierence eisting between each sub arm makes that it is impossible to adopt velocity control scheme or servo controller. Thereore, the concept o our solution is to ind an improved orce redundant mechanism which satisies a set o design conditions as ollow. (A) Necessary conditions: (A.1) The stated problem should be solved. It means the velocities o joints are independent, that is q 1 and q 2 are not required to be the same. (A.2) When all the actuators stop, the mechanism including the top plate and the internal motion, should stop. (B) Target conditions: The improved orce redundant mechanism keeps the novel perormances o orce redundant mechanism, that is the relationship, which holds between artesian space orce and joint space orce, should obey the orm shown in (6). B. Problem Solution In order to ind a mechanism which satisies above design requirements, an idea o adding one internal kinematic redundant motion θ between sub arms and End-eector comes into our mind, and the modiied orce redundant parallel mechanism is proposed in Fig.2. In this mechanism, required motion ẋ is still the top plate output velocity along ais. The procedure o calculating the kinematics is the same as the one shown in the previous subsection. As shown in Fig.2, there are still two routes to calculate the connection point velocity. In this research concept, there 5899

3 are no change o sub arms but adding internal kinematical redundancy on the connection between the top plate and sub arms. Thereore, the relationship o the irst route between connection point velocity and joint velocity is the same as (2). However, the relationship between connection point velocity and top plate velocity is ṡ 1 = ẋ R θ (7) ṡ 2 = ẋ + R θ (8) where R is the radius o the rotated round plate. Equations (7) and (8) can be written compactly as: ṡ = J T, J ẋ θ T = J To J TI where J To = 11 ] T R 2 1 is the Jacobian matri which relates artesian velocities o top plate ẋ to the velocity at the connection point, and J TI = RR ] T R 2 1 is the Jacobian matri which relates internal motion velocity θ to the velocity at the connection point respectively. ombining (2) and (9) reduces the kinematics as ollow: ] 1 q = J, J = J S J T (10) (9) ẋ θ ], is the Jacobian matri transorming rom 1 R where J = 1 R top plate velocity and internal velocity to joint velocity. Deine J o as the Jacobian matri which relates artesian velocities o top plate ẋ to the active joints velocity, and deine J I as the Jacobian matri which relates internal motion velocity θ to the active joints velocity. (10) can be written as q = J o ẋ + J I θ, J = J o J I (11) where J o = J 1 S J To R 2 1, J I = J 1 S J TI R 2 1 Hence the static orce relationship o the improved orce redundant is as ollow: =( J) T A1 (12) τ I A2 where τ I is the internal orce and is generalized orce at the top plate. Generalized orce can be derived rom (12), that is = J T o A = 11 ] = + A2 (13) A2 Now, we evaluate speciications o the improved simple orce redundant mechanism as below: (a) Necessary conditions: (a1) From (10), we know J is a regular matri when R 0, then q 1 and q 2 can have dierent values because o the introduced internal kinematic redundant motion θ. Then, (A.1) is solved. (a2) And transposing both sides o (10), we know the mechanism stops ( ẋ = θ = 0 ) when q 1 = q 2 = 0. Then, (A.2) is solved. What is more, the velocity dierence among sub arms produces internal motion but has no inluence on the required motion ẋ. (b) Target conditions: By comparing (13) with (6), we get that the top plate driving orce o both o them is the summation o joint driving orces. This static orce relationship shows that the novel perormances o orce redundant mechanism keeps. Then, (B) is satisied. Thereore, the orce redundant mechanism with kinematical redundancy has the merits summarized as below: 1) Dierences among the positions o the sub arms are allowed; 2) Each sub arms can be controlled without considering internal orces during control; 3) Fast motion can be epected. III. INTRODUING KINEMATIAL REDUNDANY INTO FORE REDUNDANT MANIPULATOR In this section, the general theory o introducing kinematical redundancy into orce redundant manipulator is described, and basic equations are proposed. Let us consider a general orce redundant parallel manipulator with m dimensional task space and n A actuations (n A > m), shown in Fig.3. In the igure, is the base coordinate rame. is the object coordinate rame ied on the top plate. Fig. 3. Active Joint Passive Joint n > m A n A: Number o active joints m : Dimension o task space The i-th sub-arm A general orce redundant parallel manipulator. Let r R m be the position and orientation o top plate. Denote q A R n A as the active joint variable o the general orce redundant parallel manipulator. Let n r indicates the number o redundant actuators and it is given by n r = n A m (14) A. Force Redundant Parallel Manipulator without Kinematic Redundancy The active joint velocity is related to the end-eector velocity in artesian space by the Jacobian matri J A R n A m q A = J A (15) 5900

4 From this equation, we can clearly know that active joint velocities are linearly dependent by using the linear algebra theory because o the mathematical relationship n A > m. Thereore, this parallel orce redundant mechanism has the disadvantage o producing big internal orces i we adopt velocity control or servo motors when designing the control scheme or real mechanism. Then the static orce relationship is F = J A T τ (16) where τ R m is the orce vector in joint space, and F R n A is the orce vector in cartesian space. As J A T R m n A and rank J A T min(m,n A ). Note that n A > m. Hence, we can get the rank J A T m. On the other hand, the dimensions o τ is n A. Thereore, by using null space o matri J A T, we proved and conirmed that this kind o parallel mechanism has orce redundancy. B. Parallel Force Redundant Manipulator with Kinematic Redundancy Based on the concept o the idea, in order to avoid the problem in the orce redundant mechanism when applying velocity control or servo motor to achieve high accuracy, introducing internal kinematic redundant motion is proposed. The number o added internal kinematical redundancy is assumed as n I. The added internal redundant motion variable α R n I (α is independent o top plate variables ), is introduced to make: q A = J A, J α A = J Ao J AI R n A (m+n I ) (17) where J A is the Jacobian matri transorming rom top plate velocity and internal velocity to active joint velocity. J Ao R na m is Jacobian matri transorming rom top plate motion velocity to active joint velocity. J AI R n A n I is the Jacobian matri transorming rom internal redundant motion velocity α to active joint velocity. What is more, this improved orce redundant mechanism should satisy the ollowing required conditions: (A) Necessary conditions: (A.1) The velocity incoincidence among sub arms is allowed. In other words, the problem o linearly dependent active joint velocities in (15) should be avoided. (A.2) When all the actuators stop, the mechanism should stop ( = 0 and α = 0 ), that is J A should be a ull rank one, i.e. The improved one keeps the novel orce relationship o orce redundant mechanism. However, it should be noted that the design o the improved mechanism should be done careully to avoid joint backlash. IV. STRUTURAL DESIGN OF MEHANISM In this section, Mechanism NINJA, shown in Fig.4, will be modiied by adding internal kinematical redundancy between sub arms and top plate. Fig. 4. High speed mechanism NINJA. A. Problem Statement o NINJA The parallel mechanism NINJA, which is designed to achieve an acceleration rate over 100 G] (G: gravitational acceleration), has our sub-arms, which are connected to a common top plate by boll joints. It is designed to be lightweight by arranging actuators on a base. Each sub-arm has two actuated joints and a passive joint. Thereore, NINJA has two degrees o orce redundancy. The simpliied structure o NINJA itsel is shown in Fig.5. ombining (18) and (14) yields: rank J A = n A = m + n I (18) n I = n r (19) Hence, rom (19), another equivalent condition, to make the Jacobian matri J A to be a ull rank square matri, is to let the number o added kinematic redundancy n I be equal to the number o redundant actuators n r. (B) Target condition: Fig. 5. Structure o NINJA. Kinematic relationship o the ith sub-arm structure is shown Fig.6. The nomenclatures are shown as ollows: : Base coordinate rame : top plate coordinate rame 5901

5 i :Thei-th ball joint Σ SJi : Sub arm coordinate rame whose origin is located at the ball joint connection between the i-th sub arm and top plate, i=1, 2, 3, 4 Σ STJi : top plate coordinate whose origin is the center o ball joint connection between the i-th sub arm and top plate B P SJi R 3 : Position o Σ SJi with respect to B P STJi R 3 : Position o Σ STJi with respect to T P STJi R 3 : Position o Σ STJi with respect to B P T = r p = y z ] T R 3 : Position o with respect to φ = ϕ θ ψ ] T R 3 : Orientation (roll-pitch-yaw) o with respect to B R T = RPY(ϕ,θ,ψ)=Rot(z,ϕ)Rot(y,θ)Rot(,ψ) R 3 3 : Roll-pitch-yaw transormation rom coordinates to top plate coordinates ω = T ω ω y ω z R 3 : Angular velocity o top plate = T p ω T ] T R 6 : Velocity o top plate q ia = T: q i1 q i2 Active joint o the ith sub arm. (qi1 and q i2 are active joints) q ip : Passive joint o the ith sub arm. (q i3 is the passive joint) q i = q T ] T: ia q ip Joint variable o the ith sub arm τ i = T: τ i1 τ i2 Joint driving orce o the ith sub arm τ = τ1 T ] T τt 2 τt 3 τt 4 R 8 : Joint driving orce o mechanism F R 6 : Generalized vector o artesian orce at top plate. q i3 Fig. 6. P SJi i PΤ T P STJi q i1 The i-th sub-arm Top plate Σ SJiΣ ( i + 1) STJi Active Joint Passive Joint Kinematic relationship o NINJA. q i2 The angular velocity o the top plate can be computed as ω = J RPY φ (24) 0 sinϕ cosϕ cosθ where J RPY = 0 cosϕ sinϕ cosθ 1 0 sinθ Applying (24) to (23), we obtain J Si q i = J Ti (25) ] 1 where J Ti = Jˆ I3 3 0 Ti 0 J RPY Hence the relationship between q i and is q i = J i (26) where J i = J Si 1 J Ti = J T ia JT ip ] T and is a unction o rp and φ, J ia R 2 6 and J ip R 1 6 are the Jacobian matri corresponding to active joint velocity q ia and passive joint velocity q ip respectively. Thereore, velocity relationship between active joints o the mechanism and the top plate is epressed compactly as q A = J A (27) where J A = J T 1A JT 2A JT 3A JT 4A ] T R 8 6 Thus the static relationship between the joint orce τ and the generalized orce F developed at the top plate is given by F = J A T τ (28) omparing (27) to (15) previously discussed, we can know that the velocities are not independent. Thereore, with the purpose o avoiding this problem, the theory o adding internal kinematic redundancy between sub-arms and top plate is applied into improving NINJA. B. Solution o NINJA - - Structural Design o New Top Plate From the origin o base coordinates to the origin o the ball joint connection point between the i-th sub arm and top plate, there are two equivalent position vectors, i.e. B P SJi = B P STJi (20) The vector o B P SJi is a unction o joint variable, i.e. B P SJi = i (q i ) (21) While B P STJi is represented as by vector addition: B P STJi = B P T + B R T T P STJi = g i (,y,z,ϕ,θ,ψ) (22) Here T P STJi is constant in the i-th closed kinematic chain. Substituting (21) and (22) into (20), and dierentiating both sides with respect to time: J Si q i = ˆ J Ti T p φ T ] T (23) Fig. 7. Structure o the modiied NINJA. By ollowing the design guidelines and taking account o the requirements discussed in the previous section, mechanism NINJA is to be modiied by designing a new top plate without changing sub arms. It means we are not adding kinematical redundancy on sub arm but on the connection points between top plate and sub arms. According to (19), the new top plate should have 2DOF as NINJA has two 5902

6 degrees o orce redundancy. What is more, new NINJA ater adopting the new top plate should obey necessary conditions and target condition presented. On the basis o these requirements, the prototype o the modiied NINJA is proposed. The simpliied structure o the new prototype is shown in Fig.7. Fig.8 shows the structure o the new top plate. These balls in the igure stand or the ball joints connecting with the subarms. P SJi i Σ SJi Σ STJi The i-th sub-arm T P STJi PΤ B A d e d e ( i + 1) o T Fig. 10. Kinematic relationship o the modiied NINJA. i Fig. 8. Fig. 9. Schematic drawing o the new top plate. : Revolute Joint : ylindric Joint o T Symmetrical and duality structure. A ( i + 1) It is a symmetrical and duality mechanism as the structure simpliied in Fig.9, and thereore it can average the position incoincidence. These proprieties are shown by e (or d, which is used to describe the relative translation motion along the vertical ais o this new top plate) and. denotes the relative rotation motion along the vertical ais o this new top plate. As a result, this top plate itsel has two internal motions which are translation and rotation along the vertical ais o this new top plate. Thereore, the internal motion is α = d ] T. The kinematic relationship o the i-th sub arm is described in Fig.10. In the kinematic equation (20), the vector o B P SJi has no change as all the arms is not changed. However, the vector o B P STJi in (20) has to be modiied as T P STJi is not a constant B vector any more but a unction o α ater adopting this new top plate. Hence, B P STJi = B P T + B R T T P STJi = g i (,y,z,ϕ,θ,ψ,,d) (29) Substituting (21) and (29) into (20), and dierentiating both sides with respect to time: J Si q i = J ˆ Ti T p φ T α T ] T (30) and premultiply both sides by J 1 Si,wehave q i = J i (31) α where J i = J Si 1 J ] T T Ti = J iat J ip and is a unction o rp, φ and α. J Ti = J ˆ Ti diag(i 3 3,J RPY,I 2 2 )] 1. J ia R 2 8 and J ip R 1 8 are the Jacobian matri corresponding to active joint velocity q ia and passive joint velocity q ip respectively. Hence the active joint velocities o the ith sub arm is: q ia = J ia (32) α where J ia = J iao J iai. JiAo R 2 6 is the Jacobian matri which relates artesian velocities o top plate to the active joints velocity, and J iai R 2 2 relates internal motion velocity α to the active joints velocity. ombining (32) together, the active joint velocities o the modiied NINJA is written compactly as q A = J A (33) α where J A = J Ao J AI ] R 8 8, J Ao = J 1Ao T J 2Ao T J 3Ao T J 4Ao T ] T R 8 6, J AI = J 1AI T J 2AI T J 3AI T J 4AI T ] T R 8 2. Then the static relationship is given by F = J Ao T τ (34) 5903

7 It is shown in (34) that the modiied NINJA keeps the orce relationship o orce redundant parallel mechanism. V. PROTOTYPE The prototype o the new top plate is shown in Fig.11 (A is the old top plate and B is the new one). Ater installing the new top plate, the prototype o the modiied NINJA is shown in Fig.12. Designing the control scheme and eperimental operation are the net works. Fig. 11. Fig. 12. Old top plate and Prototype o the new one. Prototype o Parallel Mechanism, Modiied NINJA. VI. ONLUSION With the purpose o improving the accuracy perormance in position control o orce redundant parallel mechanisms, an idea o introducing kinematical redundancy into parallel mechanisms with orce redundancy is adopted. The major results obtained in this paper are summarized as ollows. 1) The basic concept on re-design o orce redundant parallel mechanisms with kinematical redundancy is presented by using a simple eample. 2) The design procedure to realize the proposed concept is developed. 3) The proposed design procedures is illustrated by implementing it or the re-design o the eisting mechanism NINJA. A new prototype o NINJA has been designed. Future work includes implementation o a control scheme using velocity controller or the new prototype and eperimental evaluation o the perormance o the proposed mechanism. REFERENES 1] O. Masamitsu, H. Toyomi, and W. Mitsuhito, High speed and high accuracy XY stage or electronic assembly, in Proc. 4th IEEE/HMT European Int. Electron. Manuacturing Technol. Symp., Neuilly sur Seine, France, 1988, pp ] Y. Koseki. et al.; Design and Accuracy Evaluation o High-Speed and High Precision Parallel Mechanism, in Proc. IEEE Int. on. on Robotics and Automation, 1998, ] Z.Y. hu, et al.; Research o 2-DOF planar parallel high speed/high accuracy robot, Proc. 5th World ongress on Intelligent ontrol and Automation, pp , ] B. Sprenger, Design o a high speed and high precision 3 DOF linear direct drive, in Proc. 1st IEEE/ASME Int. on. Advanced Intelligent Mechatronics, Tokyo, Japan, 1997, pp ] J.P, Merlet. Parallel robots. London: Kluwer Academic Publishers; ] L. W. Tsai, Robot Analysis: The Mechanics o Serial and Parallel Manipulators. New York, USA: John Wiley & Sons, ] R. lavel, DELTA, a ast robot with parallel geometry, Proc. International Symposium on Industrial Robots, pp ] F. Pierrot, M. Uchiyama, P. Dauchez and A. Fournier, A New Design o a 6-DOF Parallel Robot, J. o Robotics and Mechatronics,vol.2, No. 4, pp , ] S. Kawamura, W. hoe, S. Tanak, S.R. Pandian, Development o an Ultrahigh Speed Robot FALON using Wire Driven Systems, IEEE Int. on. on Robotics and Automation, Nagoya, Aichi, Japan, 1995, ] J. Matsuyama, High Speed and High Accuracy Robot with losed- Loop Mechanism, 30th ISR, Vol.1, pp , ]. Gosselin and J. Angeles. Singularity analysis o closed-loop kinematic chains, IEEE Transaction on Robotics and Automation, 6(3): , ] G F.Liu, Y L. Wu, Z X. Li. Analysis and control o redundant parallel manipulators, in Proc. IEEE Int. on. on Robotics and Automation, Seoul, ] B. Dasgupta and T. S. Mruthyunjaya, Force Redundancy in Parallel Manipulators: Theoretical and Practical Issues, Mechanism and Machine Theory, Vol. 33, No. 8, (1998). 14] S. Kock and W. Schumacher, A parallel -y manipulator with actuation redundancy or high-speed and active-stiness applications, in Proc IEEE Int. on. Robotics Automation, vol. 3, pp ] Y.Nakamura, Advanced Robotics: Redundancy and Optimization, Addison-Welsey, Reading, MA, ] K.Nagai, Z.Y.Liu, Introducing Kinematical Redundancy into Parallel Mechanism with Force Redundancy, Proceedings o the 2008 IEEE/ASME International onerence on Advanced Intelligent Mechatronics, pp , ] H. heng, Y. K. Yiu, Z X. Li. Dynamics and ontrol o redundantly actuated parallel manipulators, IEEE/ASME Transactions on mechatronics. Vol.8, NO.4. pp ] S. Kock and W. Schumacher, ontrol o a ast parallel robot with a redundant chain and gearboes: Eperimental results, in Proc IEEE Int. on. Robotics and Automation, vol. 2, pp ] K.Nagai, M. Matsumoto, K. Kimura, B. Masuhara, Development o parallel manipulator NINJA with ultra-high-acceleration,in Proc.o IRA,2003,pp