Dynamics of Hexapods with Fixed-Length Legs 263 where p m and ∗ p m are the mass of the platform and the mass of the platform counterweight, i m and ∗ i m are the mass of the legs and the mass of the legs counterweights, a m and ∗ a m are the mass of the pantograph and the mass of the of the pantograph counterweight. In this case, the global center of the mass of the manipulator is written as 66 p pppaaaa ii ii i1 i1 Mm m m m m m ∗ ∗∗∗ ∗∗ == =++++ + ∑∑ rr r r r r r (83) where p r and p ∗ r are the platform center of the mass and the platform counterweight position, i r and i ∗ r are the legs center of the mass and the legs counterweight position, a r and a ∗ r are the pantograph center of the mass and the pantograph counterweight position. From Figs. 9-10, vectors p r , p ∗ r , a r , a ∗ r , i r and i ∗ r can be derived and substituted into eq.(83), yielding () ()( ) ()( ) () aa pp aa 6 gi ii iii i i1 6 gi ii iii i i1 ll Mm m m m l m l l m l ∗ ∗∗ ∗ = ∗ ∗ = ⎛⎞⎛⎞ = +⋅ + +⋅ + + + ⎜⎟⎜⎟ ⎜⎟⎜⎟ ⎝⎠⎝⎠ ⎡⎤ ++⋅−+⋅−−+ ⎢⎥ ⎣⎦ ⎡⎤ ++⋅−+⋅−− ⎢⎥ ⎢⎥ ⎣⎦ ∑ ∑ rhQg hQg h h hh hQp hQp b s hQp hQp b s (84) where, l a is the center of mass of the pantograph with respect to the fixed frame, l a * is the pantograph counterweight position with respect to the fixed frame, l gi is the length of the leg counterweight link, l i is the length of the leg, s i can be written, for i=1, ,6, as ρ = ⋅ ˆ iii ss (85) In concise form, eq.(84) can be expressed as = =++ + ∑ 6 150 ii i1 MA AhQB s Ar (86) where ∗ ∗ ∗∗ ∗ == ⎛⎞ ⎛⎞ ⎜⎟ =++ + + − + − ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ 66 gi gi aa 1ppa a i i ii i1 i1 ll ll Ammm m m1 m1 llhh (87) ∗ ∗∗ ∗ == ⎛⎞ ⎛⎞ ⎜⎟ =+ + −+ − ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ∑∑ 66 gi gi pp ii ii ii i1 i1 ll mm m1 m1 ll Bgg p p (88) 5 gi gi ii i ii ll Am m ll ∗ ∗ =+ , i=1, ,6 (89) Parallel Manipulators, Towards New Applications 264 = = ∑ 6 5ii i1 A 0 Ab (90) The conditions for static balancing can be given, for i =1, ,6, as follows = === 50 0, 0, 0, 0 1i AABA (91) From conditions = 5 0 i A , for i=1, ,6, one can obtain ∗ ∗ + = 0 gi gi ii ii ll mm ll (92) From eq. (92), for i=1, ,6, the following is obtained ∗ ∗ =− igi gi i ml l m (93) By condition = 0 1 A , i.e., 0 6 aa pp ii a a i1 ll mm (mm)m m ∗ ∗∗ ∗ = + ++++= ∑ hh (94) one can obtain () = ⎛⎞ =− + + + + ⎜⎟ ⎜⎟ ⎝⎠ ∑ * * 6 ** a appiia a i1 l lmmmmm m h h (95) Finally, condition B= 0 leads to the following () ∗∗ ∗ = + ++= ∑ 6 pp iii i1 mm mmgg p0 (96) Eq.(96) shows that the static balancing can be achieved by fixing the global center of the mass of the moving platform, that of the legs and their counterweights at the same position, O'. In order to obtain it, the platform counterweight should be placed in the position: () ∗ ∗ = ∗ ++ = ∑ 6 p iii i1 p mmm m gp g (97) Simulation is carried out to demonstrate the proposed method. The results are shown in Figs. 13-14, from which it can be seen that the centre of mass of the robot is non-stationary for non balanced case, while it is fixed for the balanced case. After static balancing the global mass of the device increases by ∗ ∗∗ = Δ=+++ ∑ 6 p aa i i1 M mmm m (98) Dynamics of Hexapods with Fixed-Length Legs 265 The negative effect for the dynamic performance by the increasing global mass can be reduced by optimum design of the pantograph. A graph can be arranged to provide such help. Fig. 15 shows the ratios, + Δ M M M , + * ii i II I , + * aa a II I , (99) which vary respect to the ratio r a * / h and l gi * / l gi and where I i is the moment of inertia of the leg, I i * is the moment of inertia of the leg counterweight whit respect of P i , I a is the moment of inertia of the moving platform and I a * is the moment of inertia of the pantograph counterweight with respect of O. It should be noted that with a suitable design it is possible to reduce M Δ at the same time, it may increase I i and I a . The effect of gravity compensation on the dynamic performances was studied in detail in (Xi, 1999). Fig. 13. Mobile center of mass Hexapod Fig. 14. Fixed center of mass of Balanced Hexapod Parallel Manipulators, Towards New Applications 266 Figure 15. Graph for optimum design Input Mobile platform mass [kg] short side [mm] long side [mm] 8 200 800 Fixed platform 1 mass [kg] short side [mm] long side [mm] / 100 400 Fixed platform 2 mass [kg] short side [mm] long side [mm] / 250 1000 leg mass [kg] l i [mm] l gi [mm] 0.5 750 375 Pantograph mass [kg] side length [mm] r a [mm] 3 100 0 Output m a * [kg] m i * [kg] 17 1 Table 3. Geometric and inertial parameters Dynamics of Hexapods with Fixed-Length Legs 267 7. Conclusion In this chapter, the inverse dynamics of hexapods with fixed-length legs is analyzed using the natural orthogonal complement method, with considering the mass of the moving platform and those of the legs. A complete kinematics model is developed, which leads to an explicit expression for the twist-mapping matrix. Based on that, the inverse dynamics equations are derived that can be used to compute the required applied actuator forces for the given movement of the moving platform. The developed method has been implemented and demonstrated by simulation. Successively, the static balancing of hexapods is addressed. The expression of the global center of mass is derived, based on which a set of static balancing equations has been obtained. It is shown that this type of parallel mechanism cannot be statically balanced by counterweights because prismatic joints do not have a fixed point to pivot as revolute joints. A new design is proposed to connect the centre of the moving platform to that of the fixed platform by a pantograph. The conditions for static balancing are derived. This mechanism is able to release the actuated joints from the weight of the moving legs for any configurations of the robot. In the future research the leg inertia will be include for modeling the dynamics of the hexapod for high-speed applications. 8. References Angeles, J. & Lee, S. (1988). The Formulation of Dynamical Equations of Holonomic Mechanical Systems Using a Natural Orthogonal Complement, ASME J. Applied Mechanics , Vol. 55, pp. 243-244, ISSN: 0021-8936 Angeles, J. & Lee, S. (1989). The Modeling of Holonomic Mechanical Systems Using a Natural Orthogonal Complement, Trans. Canadian Society of Mechanical Engineering, Vol. 13, No. 4, pp. 81-89, ISSN: 0315-8977 Angeles, J. & Ma, O. (1988). Dynamic Simulation of n-axis Serial Robotic Manipulators Using a Natural Orthogonal Complement, The International Journal of Robotics Research , Vol. 7, No. 5, pp. 32-47, ISSN: 0278-3649 Do, W. Q. D. & Yang, D. C. H. (1988). Inverse Dynamics and Simulation of a Platform Type of Robot, The International Journal of Robotics Research, Vol. 5, No. 3, pp. 209-227, ISSN: 0278-3649 Fichter, E. F. (1986). A Stewart Platform-based Manipulator: General Theory and Practical Construction, The International Journal of Robotics Research, Vol. 5, No. 2, pp. 157-182, ISSN: 0278-3649 Fijany, A., & Bejezy, A. K., (1991). Parallel Computation of Manipulator Inverse Dynamics, Journal of Robotic Systems, Vol. 8, No. 5, pp. 599-635, ISSN: 0741-2223 Geng, Z.; Haynes, L. S.; Lee, T. D. & Carroll, R. L. (1992). On the Dynamic and Kinematic Analysis of a Class of Stewart Platforms, Robotics and Autonomous Systems, Vol. 9, No. 4, pp. 237-254, ISSN: 0921-8890 Gosselin, C. M. & Wang, J. (1998). On the design of gravity-compensated six-degree-of- freedom parallel mechanisms, Proceedings of IEEE International Conference on Robotics and Automation , Leuven, Belgium, May 1998, ISBN: 0-7803-4300-X Hashimoto, K. and Kimura, H., (1989). A New Parallel Algorithm for Inverse Dynamics, The International Journal of Robotics Research , Vol. 8, No. 1, pp. 63-76, ISSN: 0278-3649 Hervé, J. M. (1986). Device for counter-balancing the forces due to gravity in a robot arm, United States Patent, 4,620,829 Parallel Manipulators, Towards New Applications 268 Honegger, M.; Codourey, A. & Burdet, E. (1997). Adaptive Control of the Hexaglide a 6 DOF Parallel Manipulator, Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, NM, April 1997, ISBN: 0-7803-3612-7 Kazerooni, H. & Kim, S. (1990). A new architecture for direct drive robot, Proceedings of the ASME Mechanism Conference, Vol.DE-25, pp. 21-28, Sept. 16-19, 1990, Chicago, IL Nathan, R. H., (1985). A constant force generation mechanism, ASME Journal of Mechanism, Transmissions, and Automation in Design , Vol. 107, No. 4, pp 508-512, ISSN: 0738-0666 Pritschow, G., & Wurst, K.–H., (1997). Systematic Design of Hexapods and Other Parallel Link Systems, Annals of the CIRP, Vol. 46/1, pp. 291-295, Elsevier, ISSN: 0007-8506 Saha, K.S. & Angeles, J. (1991). Dynamics of Nonholonomic Mechanical Systems Using a Natural Orthogonal Complement, ASME Journal of Applied Mechanics, Vol. 58, pp.238-243, ISSN: 0021-8936 Streit, D.A. & Gilmore, B.J. (1989). Perfect spring equilibrator for rotatable bodies, ASME Journal of Mechanisms, Transmissions, and Automation in Design , Vol. 111, No. 4, pp. 451-458, ISSN: 0738-0666 Streit, D.A. & Shin, E. (1990). Equilibrators for planar linkage. Proceedings of the ASME Mechanism Conference , Vol. DE-25, pp. 21-28, Cincinnati, OH, , May 1990, Chicago, ISBN: 0-8186-9061-5 Sugimoto, K. (1987). Kinematic and Dynamic Analysis of Parallel Manipulators by Means of Motor Algebra, ASME Journal Mechanisms, Transmissions, and Automation in Design, Vol. 109, pp. 3-7, ISSN: 0738-0666 Susuki, M.; Watanabe K.; Shibukawa, T.; Tooyama, T. & Hattori, K. (1997). Development of Milling Machine with Parallel Mechanism, Toyota Technical Review, Vol. 47, No. 1, pp. 125-130 Tlusty, J.; Ziegert, J. & Ridgeway, S. (1999). Fundamental Comparison of the Use of Serial and Parallel Kinematics for Machine Tools, Annals of the CIRP, Vol. 48, pp. 351-356, Elsevier, ISSN: 0007-8506 Tsai, L. W. (2000). Solving the Inverse Dynamics of a Stewart-Gough Manipulator by the Principle of Virtual Work, ASME Journal of Mechanical Design, Vol. 122, pp. 3-9, ISSN: 1050-0472 Ulrich, N. & Kumar, V. (1991). Passive mechanical gravity compensation for robot manipulators, Proceedings of the IEEE International Conference on Robotics and Automation , pp. 1536-1541, Sacramento, CA, USA, April 1991, ISBN: 0-8186-2163-X Walsh, G.J.; Strei, D.A & Gilmore, B.J. (1991). Spatial spring equilibrator theory, Mechanism and Machine Theory , Vol. 26, No 2, pp. 155-170, ISSN: 0094-114X Wang, J. & Gosselin, C. M. (2000). Static balancing of spatial four-degree-of freedom parallel mechanisms, Mechanism and Machine Theory, Vol. 35, pp. 563-592, ISSN: 0094-114X Wang, J. & Gosselin, C.M. (1999). Static balancing of spatial three degree freedom parallel mechanism, Mechanism and Machine Theory, Vol. 34, pp. 437-452, ISSN 0094-114X Xi F.; Russo, A. & Sinatra, R., (2005). Static Balancing of parallel robots, Mechanism and Machine Theory , Vol. 40, No 2, pp. 131-258, ISSN: 0094-114X Xi, F. & Sinatra, R. (1997). Effect of Dynamic Balancing on Four-bar Linkage Vibrations, Mechanism and Machine Theory, Vol. 32, No. 6, pp.715-728, ISSN: 0094-114X Xi, F. & Sinatra, R. (2002). Inverse Dynamics of Hexapods using the Natural Orthogonal Complement Methods, Journal of Manufacturing Systems, Vol. 21, No 2, pp.73-82, ISSN: 0258-6125 Xi, F. (1999). Dynamic Balancing of Hexapods for High-speed Applications, Robotica, Vol. 17, pp. 335-342, ISSN: 0263-5747 Zanganesh, K. E.; Sinatra, R. & Angeles J. (1997). Dynamics of a Six-Degree-of-Freedom Parallel Manipulator with Revolute Legs, Robotica, Vol. 15, pp. 385-394, ISSN: 0263- 5747. 13 Cartesian Parallel Manipulator Modeling, Control and Simulation Ayssam Elkady 1 , Galal Elkobrosy 2 , Sarwat Hanna 2 and Tarek Sobh 1 University of Bridgeport 1 , Alexandria University 2 , USA 1 , Egypt 2 1. Introduction Parallel manipulators are robotic devices that differ from the more traditional serial robotic manipulators by their kinematic structure. Parallel manipulators are composed of multiple closed kinematic loops. Typically, these kinematic loops are formed by two or more kinematic chains that connect a moving platform to a base, where one joint in the chain is actuated and the other joints are passive. This kinematic structure allows parallel manipulators to be driven by actuators positioned on or near the base of the manipulator. In contrast, serial manipulators do not have closed kinematic loops and are usually actuated at each joint along the serial linkage. Accordingly, the actuators that are located at each joint along the serial linkage can account for a significant portion of the loading experienced by the manipulator, whereas the links of a parallel manipulator generally need not carry the load of the actuators. This allows the parallel manipulator links to be made lighter than the links of an analogous serial manipulator. The most noticeable interesting features of parallel mechanisms being: • High payload capacity. • High throughput movements (high accelerations). • High mechanical rigidity. • Low moving mass. • Simple mechanical construction. • Actuators can be located on the base. However, the most noticeable disadvantages being: • They have smaller workspaces than serial manipulators of similar size. • Singularities within working volume. • High coupling between the moving kinematic chains. 1.1 Prior work Among different types of parallel manipulators, the Gough-Stewart platform has attracted most attention because it has six degrees of freedom (DOF). It was originally designed by (Stewart, 1965). Generally, this manipulator has six limbs. Each one is connected to both the base and the moving platform by spherical joints located at each end of the limb. Parallel Manipulators, Towards New Applications 270 Actuation of the platform is typically accomplished by changing the lengths of the limbs. Although these six-limbed manipulators offer good rigidity, simple inverse kinematics, and high payload capacity, their forward kinematics are difficult to solve, position and orientation of the moving platform are coupled and precise spherical joints are difficult to manufacture at low cost. To overcome the above shortcomings, parallel manipulators with fewer than six degrees of freedom have been investigated. For examples, (Ceccarelli, 1997) proposed a 3-DOF parallel manipulator (called CaPaMan) in which each limb is made up of a planar parallelogram, a prismatic joint, and a ball joint. But these manipulators have coupled motion between the position and orientation of the end-effector. The 3-RRR (Revolute Revolute Revolute) spherical manipulator was studied in detail by (Gosselin & Angeles, 1989). Several spatial parallel manipulators with a rotational moving platform, called rotational parallel manipulators (RPMs), were proposed (Di Gregorio, 2001), (Karouia & Herve, 2000) and (Vischer & Clavel, 2000). (Clavel, 1988) at the Swiss Federal Institute of Technology designed a 3-DOF parallel manipulator that does not suffer from the first two of the listed disadvantages of the Stewart manipulator. Closed-form solutions for both the inverse and forward kinematics were developed for the DELTA robot (Gosselin & Angeles, 1989). The DELTA robot has only translational degrees of freedom. Additionally, the position and orientation of the moving platform are uncoupled in the DELTA design. However, the DELTA robot construction does employ spherical joints. (Tsai, 1996) presented the design of a spatial 3-UPU (Universal Prismatic Universal) manipulator and pointed out the conditions that lead to pure translational motion and its kinematics was studied further by (Di- Gregorio & Parenti-Castelli, 1998). (Tsai, 1996) and (Tsai et al., 1996) designed a 3-DOF TPM (Translational Parallel Manipulator) that employs only revolute joints and planar parallelograms. (Tsai & Joshi, 2002) analyzed the kinematics of four TPMs for use in hybrid kinematic machines. (Carricato & Parenti-Castelli, 2001) developed a family of 3-DOF TPMs. (Fang & Tsai, 2002) presented a systematic methodology for structure synthesis 3-DOF TPMs using the theory of reciprocal screws (Kim & Tsai, 2002). Han Sung Kim and Lung-Wen Tsai (Kim & Tsai, 2002) presented a parallel manipulator called CPM (figure 1) that employs only revolute and prismatic joints to achieve translational motion of the moving platform. They described its kinematic architecture and discussed two actuation methods. For the rotary actuation method, the inverse kinematics provides two solutions per limb, and the forward kinematics leads to an eighth-degree polynomial. Also, the rotary actuation method results in many singular points within the workspace. On the other hand, for the linear actuation method, there exists a one-to-one correspondence between the input and output displacements of the manipulator. Also, they discussed the effect of misalignment of the linear actuators on the motion of the moving platform. They suggested a method to maximize the stiffness to minimize the deflection at the joints caused by the bending moment because each limb structure is exposed to a bending moment induced by the external force exerted on the end-effector. 2. Manipulator description and kinematics 2.1 Manipulator structure The Cartesian Parallel Manipulator, shown in figure 1, consists of a moving platform that is connected to a fixed base by three limbs. Each limb is made up of one prismatic and three revolute joints and all joint axes are parallel to one another. Cartesian Parallel Manipulator Modeling, Control and Simulation 271 Figure 1: Assembly drawing of the prototype parallel manipulator. 2.2 Kinematic structure The kinematic structure of the CPM is shown in figure 2 where a moving platform is connected to a fixed base by three PRRR (Prismatic Revolute Revolute Revolute) limbs. The origin of the fixed coordinate frame is located at point O and a reference frame XYZ is attached to the fixed base at this point. The moving platform is symbolically represented by a square whose length side is 2L defined by B 1 , B 2 , and B 3 and the fixed base is defined by three guide rods passing through A 1 , A 2 , and A 3 , respectively. The three revolute joint axes in each limb are located at points A i , M i , and B i , respectively, and are parallel to the ground- connected prismatic joint axis. Furthermore, the three prismatic joint axes, passing through point A i , for i = 1, 2, and 3, are parallel to the X, Y, and Z axes, respectively. Specifically, the first prismatic joint axis lies on the X-axis; the second prismatic joint axis lies on the Y axis; and the third prismatic joint axis is parallel to the Z axis. Point P represents the center of the moving platform. The link lengths are L 1 , and L 2 . The starting point of a prismatic joint is defined by d 0i and the sliding distance is defined by d i - d 0i for i = 1, 2, and 3. 2.3 Kinematics constraints For this analysis, the position of the end-effector is considered known, and is given by the position vector P= [x, y, z] which defines the location of P at the center of the moving platform in the XYZ coordinate frame. The inverse kinematics analysis produces a set of two joint angles for each limb (θ i1 and θ i2 for the i th limb) that define the possible postures for each limb for the given position of the moving platform. Parallel Manipulators, Towards New Applications 272 Figure 2: Spatial 3-PRRR parallel manipulator. 2.3.1 The first limb A schematic diagram of the first limb of the CPM is sketched in figure 3, and then the relationships for the first limb are written for the position P[x, y, z] in the coordinate frame XYZ. Figure 3: Description of the joint angles and link lengths for the first limb. y= L 1 cos θ 11 +L 2 cos θ 12 +L (1) z= L 1 sin θ 11 +L 2 sin θ 12 (2) 222 2111 111 (cos )(sin)LyL L zL θ θ =− − +− (3) [...]... 436 0.0027 0.0223 8507 436 3.4804 x 10- 4 0.021 7053 436 3.0256 x 10- 4 0.0182 2550.25 101 2.3469 x 10- 4 0.0161 Table 3: The performance of various controllers Figure 11 Position error of the end-effector obtained from the Simple PD Controller Figure 12: Velocity error of the end-effector obtained from the Simple PD Controller 288 Parallel Manipulators, Towards New Applications Figure 13: The actuator... z-axis is Z=0.3 meters The desired force obtained from the actuators to move the end-effector along the desired trajectory is shown in figure 10 Figure 9: End-effector path for the circular trajectory 286 Parallel Manipulators, Towards New Applications Figure 10: The desired force obtained from the actuators 5.4 Simulation results To investigate each controller’s performance, computer simulation, carried... Translational Parallel Manipulators , Proceedings of the 2001 ASME Design Engineering Technical Conferences, Pittsburgh, PA, DAC- 2103 5 Ceccarelli, M., (1997), “A New 3 D.O.F Spatial Parallel Mechanism”, Mechanism and Machine Theory, Vol 32, No 8, pp 895-902 Clavel, R., (1988), “Delta, A Fast Robot with Parallel Geometry”, Proceedings of the 18th International Symposium on Industrial Robots, pp 91 -100 Di Gregorio,... ζ where: 284 Parallel Manipulators, Towards New Applications ωn = K P , ζ= KD 2 KP (45) Since the equation 44 is linear, it is easy to choose KD and KP so that the overall system is stable and e → 0 exponentially as t→∞ Usually, if KD and KP are positive diagonal matrices, the control law 43 applied to the system 39 results in exponentially trajectory tracking It is customary in robot applications. .. coordinates 3.2 Derivation of the manipulator‘s dynamics 3.2.1 The kinetic and potential energy of the first limb Referring to figure 6, the velocities of A1 (the prismatic joint of the first limb), A2 and A3 are x , y and z The angular velocity of the rod A1 M1 is θ11 We can consider the moment of 276 Parallel Manipulators, Towards New Applications inertia of rods A1 M1, A2 M2, and A3 M3 is I = m1 L 2 m3... ) = 2Ax , =0 dt ∂x ∂x 3 d ∂L ∂L ∂f ( )− = ∑ (λi i ) + Q 4 ∂x i =1 ∂x dt ∂x (35) 280 Parallel Manipulators, Towards New Applications =F 2A x x − Γ11λ1 − Γ 21λ2 − Γ 31λ3 (36) where Fx , Fx and Fx are the forces applied by the actuator for the first, second and third limbs Γ ij is the (i, j) element of the Γ matrix 3.2 .10 Taking the derivatives of the Lagrange function with respect to Y d ∂L ∂L ( ) = 2Ay... the second PD Controller Cartesian Parallel Manipulator Modeling, Control and Simulation Figure 16: The actuator force required by the second PD Controller Figure 17: Position error of the end-effector obtained from the third PD Controller Figure 18: Velocity error of the end-effector obtained from the third PD Controller 289 290 Parallel Manipulators, Towards New Applications Figure 19: The actuator... rejection of disturbances and parameter variations Because of such limitation, model based control algorithms were proposed (Sciavicco et al., 1990) that have the potential to perform better than independent 282 Parallel Manipulators, Towards New Applications joint controllers that do not account for manipulator dynamics However, the difficulty with the model based controller is that it requires a good... difference between the total kinetic energy, T, and the total potential energy V: L= T− V L = A (x 2 + y 2 + z 2 ) + B (θ 2 + θ 2 + θ 2 ) + C (sin θ11 + cos θ21 ) + Ez 11 21 31 (20) 278 Parallel Manipulators, Towards New Applications Where: m1 m2 2 , m + m2 1 ) L1 C = 1 gL1 and E = (m1 + 2m 2 + m 3 + m 4 ) g A = [ m1 + 2m 2 + m 3 + m 4 ] , B = ( + 6 4 2 2 3.2.5 The constraint equations Differentiation... in designing and controlling motion of the robot to achieve the highest quality and quantity of work The simulation results show that the computed torque controller gives the best 292 Parallel Manipulators, Towards New Applications performance This is a result of the computed torques canceling the nonlinear components of the controlled system From the observations seen in this work, one can see the . Patent, 4,620,829 Parallel Manipulators, Towards New Applications 268 Honegger, M.; Codourey, A. & Burdet, E. (1997). Adaptive Control of the Hexaglide a 6 DOF Parallel Manipulator,. the limb. Parallel Manipulators, Towards New Applications 270 Actuation of the platform is typically accomplished by changing the lengths of the limbs. Although these six-limbed manipulators. limb for the given position of the moving platform. Parallel Manipulators, Towards New Applications 272 Figure 2: Spatial 3-PRRR parallel manipulator. 2.3.1 The first limb A schematic